a survey of community detection methods in multilayer …...nity detection methods in multilayer...

Data Mining and Knowledge Discovery (2021) 35:1–45https://doi.org/10.1007/s10618-020-00716-6

A survey of community detection methods in multilayernetworks

Xinyu Huang1 · Dongming Chen1 · Tao Ren1 · Dongqi Wang1

Received: 24 February 2020 / Accepted: 10 September 2020 / Published online: 13 October 2020© The Author(s) 2020

AbstractCommunity detection is one of the most popular researches in a variety of complexsystems, ranging from biology to sociology. In recent years, there’s an increasingfocus on the rapid development of more complicated networks, namely multilayernetworks. Communities in a single-layer network are groups of nodes that are morestrongly connected among themselves than the others, while in multilayer networks,a group of well-connected nodes are shared in multiple layers. Most traditional algo-rithms can rarely perform well on a multilayer network without modifications. Thus,in this paper, we offer overall comparisons of existing works and analyze severalrepresentative algorithms, providing a comprehensive understanding of communitydetection methods in multilayer networks. The comparison results indicate that thepromoting of algorithm efficiency and the extending for general multilayer networksare also expected in the forthcoming studies.

Keywords Community detection · Multilayer network · Temporal network ·Multiplex network · Multilevel network

Responsible editor: Tim Weninger.

B Dongming [email protected]

Xinyu [email protected]

Tao [email protected]

Dongqi [email protected]

1 Software College, Northeastern University, Shenyang 110169, China

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s10618-020-00716-6&domain=pdf

http://orcid.org/0000-0001-7863-1230

2 X. Huang et al.

1 Introduction

Network theory is an important tool for describing and analyzing complex systemsthroughout a variety of disciplines. Community structures, defined as groups of nodesthat are more densely connected than with rest of the network, are widely existed inmany real-world complex systems, such as sociology, biology, transportation systems,and so on (Newman 2018). Discovering communities in these systems has become aprimary approach to understand how network structure relates to system behaviors.As an effective technique to unveil the underlying structures, community detection hasbeen utilized in many scenarios, such as finding potential friends in social media (Zhuet al. 2017), recommending products for users (Li and Zhang 2020), analyzing socialopinions (Wang et al. 2017), and so on.

With the deepening of research, more and more scholars come to realize thatsimply uncovering communities in a single network is insufficient to analyze thestructures and system behaviors in real-life applications. Unlike the community struc-ture in single-layer networks, communities in multilayer networks are comprised ofa group of well-connected nodes in all layers. For example, individuals in social net-works may have various interactions (e.g. sending emails, participating in the sameactivity) among them (Ansari et al. 2011). As a result, the conventional studies areencountering with an essential problem of how to utilize the multiple views of thenetwork (Papalexakis et al. 2013). There are also similar scenarios with relevant nota-tions such as multiplex networks (Verbrugge 1979), multilevel networks (Wang et al.2013), network of networks (Gao et al. 2011), interdependent networks (Buldyrev et al.2010), multi-dimensional networks (Berlingerio et al. 2011c), which can be generallyregarded as multilayer networks (Kivelä et al. 2014). As more interaction informationimplied, community detection in multilayer networks has been introduced to leveragevarious relationships to get more accurate results (Liu et al. 2018).

1.1 Background

The research of complex networks oriented from graph theory, which is started fromthe “Seven Bridges of Königsberg” problem in 1736. Although naive inmany respects,this approach has been extremely successful in many real-life applications. At the endof the 20th century, by employing the graph model, the famous small-world (Wattsand Strogatz 1998) and scale-free (Barabási and Albert 1999) features were dis-covered (Newman 2018). In 2002, Girvan and Newman uncovered the communitystructure in networks (Girvan and Newman 2002), which opened up a continuousupsurge of relevant research. The so-called community structures are groups of nodesthat are more strongly or frequently connected among themselves than with the others.Therefore, community detection is proposed to find the most reasonable partitions ofa network via observed topological structures. Several conventional approaches arelisted in Table 1.

In recent years, a sea of improved algorithms based on the above approaches areproposed with fruitful results. However, with the exponential growth of data scale,community detection on large volume data has encountered a serious of problems. For

123

A survey of community detection methods in multilayer networks 3

Table1

Com

parisonof

someclassiccommun

itydetectionalgo

rithms

Nam

eParameters

Com

plexity

Strategy

GN(G

irvanandNew

man

2002

)–

O(nm2)

Rem

ovalof

edgeswith

maxim

umbetweenness

FN,C

NM

(GirvanandNew

man

2002

)–

O(n

2),O

(nlog2

n)Greedymodularity

maxim

ization

LPA

(Raghavanetal.2

007)

iO

(n)

Labelpropagation

EM

(New

man

andLeicht2

007)

k,i

O(nki

)Probabilisticmixture

models

SCAN(X

uetal.2

007)

ε,μ

O(n

)Structuralclustering

Lou

vain

(Blond

eletal.2

008)

–O

(nlogn)

Multilevelmodularity

maxim

ization

LFM

(Lancichinettietal.2

009)

αO

(n2logn)

Localoptim

ization

INFO

MAP(RosvallandBergstrom

2008

;Yiapanisetal.2

013)

τO

(m(n

+m

))Inform

ationcompression

NMF(Zhang

etal.2

007)

–O

(hkn

2)

Nonnegativ

ematrixfactorization

nisthenu

mbero

fnod

esandmdepictsthenumbero

fedges.G

Nisshortfor

Girvan–New

man

Algorith

m.FNisfrom

Clauset,N

ewman,and

Moore’swork“Finding

community

structurein

very

largenetworks”,CNM

isan

improved

versionusingheap

structureandfaster

updatin

gstrategy.L

abelPropagationAlgorith

m(i.e.L

PA)hasnearly

linear

complexity,w

here

parameter

idepictsthenumberof

iteratio

ns.P

aram

eter

αin

LFM

algorithm

controlsthescaleof

detected

communities.Inadditio

n,LFM

algorithm

can

beused

todetectoverlapp

ingcommun

ities.E

Malgo

rithm,o

rExp

ectatio

n-maxim

izationalgo

rithm,requiresparameterskandi,where

kisthenu

mberof

commun

ities,i

istheiterativ

etim

es.τ

inIN

FOMAPalgorithm

depictstheprobability

ofarandom

walk,

usually

assigned

τ=

0.5.

hin

NMFalgo

rithm

representsthenu

mberof

iteratio

nsuntil

convergence,kisthedimensionality

value

123

4 X. Huang et al.

example, some datasets are changingwith time, which brings challenges for traditionalresearch on a static graph. In other words, traditional methods are incapable of dealingwith large-scale time-varying networks.

Lancichinetti et al. (2009) raise two problems: The first one is about hierarchicalstructure, which means large communities are composed of small communities andin turn, large communities group together to form even larger ones. The second isoverlapping communities, for example, people belong to more than one community,depending on their families, friends, professions, hobbies, etc. Nodes belong to morethan one community is a pervasive scenario in networks.

Besides, the interactions among nodes are becoming more complicated than everbefore. The conventional monolayer network (i.e. single-layer network) has providedplentiful cases in which a unit in a complex system is charted into a node, and theinteractions between units are straightly represented as edges, no matter what typeor weight of the interactions are. With the development of network modeling, we arestarting to realize that the existing models cannot fully capture the details existed insome real-life problems,which even leads to incorrect descriptions of somephenomenataking place on real-world networks. Some representative problems occurred are listedas follows.

(1) Multiple interactions among social networks. There has been an increasing focuson social media such as Twitter, Facebook, and Google+, etc. People share theiropinions on daily affairs, chat with friends, or evenmake trades on these platforms.Themain problem for analyzing social networks is themultiple interactions amongindividuals. For example, relationships between two individuals may includefriendship, kinship, or schoolmates. If we regard all the relationships as undis-tinguished edges, the differences will be ignored, which is probably leading toincorrect results.

(2) Interbank trades. Online payments are replacing traditional cash payments andcredit cards, which offers a convenient lifestyle, meanwhile, providing a new wayfor financial criminals. For example, in money laundering activities (Colladon andRemondi 2017), criminals use different channels to conduct trades. If we simplyanalyze the transfer records from a single bank with a graph model, the resultmay be unconvinced. Thus, it is necessary to collect data from all possible tradechannels with a multilayer network model, in which each channel can be regardedas a layer.

(3) Urban transportation system. The study on the urban transportation system hascausedwide public concern in the last decade (Black 2018; Sultana et al. 2019).Cit-izens travel in a variety of ways such as bus, subway, tram, and so on. In analyzingthe public transportation system, the characteristics of various modes of trans-portation should be fully considered, especially for some hub stations (Zhang et al.2018b) that should be given more attention to solving traffic congestion problems.When bus routes are blocked, numerous passengers select subway alternatively,which results in crowd subway operations. Inherently, the urban transportationsystem is a multilayer network model.

By tackling the above problems with the multilayer network model (Kivelä et al.2014; Boccaletti et al. 2014), we are able to get a more reasonable result, i.e., the

123


community structures in multilayer networks benefit to identify functionally cohesivesub-units and reveal complex interactions and heterogeneous links.

1.2 Main contributions

There have been numerous attempts to address community detection problem inmultilayer networks through diverse approaches, e.g., identifying communities in tem-poral networks by modularity-maximization (Bazzi et al. 2016), where the authorsemphasize the difference between “null networks” and “null models” in modularitymaximization and discuss the effect of interlayer edges on the multilayer modularity-maximization problem. De Bacco et al. (2017) propose a generative model formultilayer networks, which can be used to aggregate layers into clusters or to compressa dataset by identifying especially relevant or redundant layers. The proposed modelis capable of incorporating community detection and link prediction for multilayernetworks, and experimental results on both synthetic and real-world datasets showsits feasibility. Analyzing multilayer networks is of great importance because manyinteresting patterns cannot be obtained by analyzing single-layer networks. That’s ourmotivation for summarizing these approaches. The contributions of this work are:

(1) Webuild a taxonomyof community detectionmethods based on various techniquesused.

(2) We provide a detailed survey of works that come under different categories.(3) The evaluation measures for community results are categorized and summarized.(4) The applications of community detection in multilayer networks are introduced,

as well as interesting directions for future works.

To the best of our knowledge, this is the latest work that provides a comprehensivesurvey on various community detection methods in multilayer networks.

1.3 Outline of the paper

The remainder is organized along the other 5 sections. In the next section, we start bypresenting the multilayer network models with several real-world datasets and givebrief comparisons on different definitions. Section 3 summarizes the existing commu-nity detection methods in multilayer networks and provides some metrics for qualityevaluation. We introduce various applications of community detection in multilayernetworks in Sect. 4, such as temporal networks, social networks, transportation sys-tems, and biological systems. Section 5 offers concluding remarks and perspectiveideas.

2 Models

As mentioned above, numerous researchers dedicated to solving the problems intheir own situations. In the 1930s, sociograms (Roethlisberger and Dickson 2003)were proposed to represent social relationships in a banking room, which contains14 individuals via 6 different types of social interactions, as shown in Fig. 1. Such

123

6 X. Huang et al.

Fig. 1 The sociograms proposed by Roethlisberger et al. contains 14 individuals via 6 different types ofsocial interactions, observed friendship ties and cliques in a factory. Position reflects the location of theirworkspace

networks are known as “multiplex networks” (Mucha et al. 2010) or “multi-relationalnetworks” (Cai et al. 2005) in which edges are categorized by their types.

In recent years, great endeavors have been made to unveil the basic mechanisms forthe generation of networkswith specific structural properties. The analysis of networkshas profound implications in very different fields, from social media analytics to biol-ogy (Newman 2018). The conventional graph model is incapable when the networkis differentiated, multipartite, integrated, and dynamic. Thus, a series of more com-plicated models are proposed, such as temporal networks (Kostakos 2009), multiplexnetworks, k-partite networks, and so on (Boccaletti et al. 2014; Kivelä et al. 2014).However, the sudden and immense explosion of research on multilayer networks hasalso led to a great deal of confusion (Ahmed et al. 2018; Farooq and Zhu 2018; Kiveläet al. 2014; Paolucci 2018). The multilayer network we focus, in this paper, is not aneural network but a mathematic model which can be utilized to represent the com-plicated network structures.

A multilayer network is a network made of multiple graphs, called layers, whichshare the same set of nodes, but differ in their edges. To distinguish the definitionof a multilayer network from a single-layer network, we intuitively compared therepresentation of a real-world dataset between the two models. Al Qaeda cell wasisolated from a safe hiding place during training and plan terrorist attacks, which

123


Fig. 2 The social relationships of 9/11 terrorists represented by a graph model. It is a single-layer network,which consists of 69 nodes and 159 edges. The size of the node represents the number of the neighbors(i.e., the node’s degree)

forced the organization to forma relatively dense social network, inwhich theHamburgbranch planned and eventually participated in the implementation of the September11 attacks (Silber 2011). The social relationships of 9/11 terrorists are represented inFig. 2.

If we classify the closeness of the social actions among the terrorists, we can obtainthree groups of links, abstracted into a three-layered network, as shown in Fig. 3.

It is obvious that the multilayer networks reveal more detailed information than themonolayer network. The multilayer network model contains nodes and edges fromdifferent layers, which represents the different frequencies (or types) of interactionsamong them.

The nodes in a multilayer network are consistent in that of graph model, whichrepresents individuals across multiple layers likewise. Specifically, some works indi-cate that the nodes in a multilayer network can be classified into different categories(e.g. bipartite network), thereby the network composed of these nodes is described asa node-colored network (Baltakiene et al. 2018; Brummitt 2014; Kivelä et al. 2014).The different colors represent different types of nodes. For example, the urban trans-portation system mainly contains buses, subway, and so on. The bus stop and subwaystation make no difference in a conventional graph model but differ in a multilayernetwork for the different manners of transportation, thereby they are printed with dif-

123

8 X. Huang et al.

Fig. 3 The 9/11 terrorists’ interactions represented by a multilayer network. L1 presents confirmed closecontact, L2 shows various recorded interactions, L3 contains potential or planed or unconfirmed interactions

Fig. 4 The transportation system in Chengdu city of China, where the left layer shows the bus lines and theright layer shows the subway system. The nodes from different layers are connected by interlayer edges ifthe corresponding stations are within 0.5km

ferent colors. Inherently, the layers’ information has covered the different node colors,i.e., the multilayer network model is a capable solution. The transportation networkof Chengdu city is plotted by Muxviz (De Domenico et al. 2015b), as shown in Fig. 4.

The edges in a multilayer network are classified into intralayer edges and interlayeredges (De Domenico et al. 2016). The intralayer edges request the two nodes of thisedge are in the same layer, while the interlayer edges (or crossed layer edges) connectnodes among different layers, as illustrated in Fig. 5.

A layer in multilayer networks is composed of a set of nodes and edges, i.e., a graphmodel. The layers are also organized into two categories:

123


Fig. 5 A three-layered toynetwork. The edges in each layerare called intralayer edges, asmarked by solid lines, thedashed lines crossed adjacentlayers represent interlayer edges

– Ordinal layers. The layers are sorted by a certain order, inwhich the interlayer edgesconnect the corresponding nodes in the adjacent layers. Take temporal networksfor example, there are numerous layers representing different snapshots, but theorder of layers is decided by the time sequence.

– Categorical layers. The layers are classified into several groups, where each grouprepresents a type of interaction.

2.1 Definitions

There are many terms for describing multilayer networks, such as multiplex network,multi-relational network, edge-colored network, node-colored network, multilevelnetwork, multi-dimensional network, independent networks, networks of networks,temporal network and so on (Boccaletti et al. 2014; Kivelä et al. 2014). Table 2 sum-marizes the main notations used throughout this paper.

As we have known, a graph is a tuple G = (V , E), where V is a set of nodesand E ⊆ V × V is the set of edges that connect pairs of nodes (Bollobás 2013). Themodel of multilayer networks is more complicated and there are mainly two kinds ofexplanations. One is summarized by Kivelä et al. (2014), described as

M = (VM , EM , V ,L), (1)

where VM ⊆ V ×L1 ×L2 × · · · ×Ld , EM ⊆ VM × VM . They employed a sequenceL, described as

L = {Lα}dα=1, (2)

123

10 X. Huang et al.

Table 2 Main symbols used in this paper

Symbol Scenario Description

R General definitions Set of real numbers

G Monolayer A graph or a monolayer network

n Monolayer The number of nodes

m Monolayer The number of edges

V Monolayer Node set in a graph or monolayer networks

E Monolayer Edge set in a graph or monolayer networks

ki Monolayer The degree of node i

Γ (i) Monolayer The neighbors of node i

Q Monolayer Modularity (Newman and Girvan 2004)

D Monolayer Modularity density (Li et al. 2008)

S Monolayer Surprise (Aldecoa and Marín 2011)

L Multilayer The total number of layers

G Multilayer A multilayer network comprised of graphs

C Multilayer Collection of interlayer edges

α Multilayer Aspect (Kivelä et al. 2014) or layer

Gα Multilayer A graph of layer α

Vα Multilayer Node set in layer α

Eα Multilayer Edge set in a layer α

L Multilayer Collection of layers

Lα Multilayer Elementary layer α

M Multilayer Multilayer network model

M Multilayer The supra-adjacency matrix

Aα Multilayer The adjacency matrix of layer α

Iαβ multilayer The interlayer edges between layer α and β

kαi Multilayer The degree of node i in layer α

QM Multilayer Modularity for multiplex networks

ρc Miscellaneous Redundancy index

δ Miscellaneous Kronecker delta

where α is called aspect, and Lα depicts an elementary layer. The layer is a product ofelementary layers, which can be represented as L1 × L2 × · · · × Ld . The illustrationof the multilayer network is given in Fig. 6.

Another model is proposed by Boccaletti et al. (2014), defined as

M = (G, C), (3)

where G = {Gα;α ∈ {1, . . . , L}} is a family of (directed or undirected weighted orunweighted) graphs Gα = (Vα, Eα), which represents layers ofM and C depicts the

123


Fig. 6 The illustrated multilayer network model proposed by Kivelä et al. As shown in the left panel,the multilayer network has a total of four nodes, so V = 1, 2, 3, 4. There are two aspects, which hascorresponding elementary-layer sets L1 = {A, B} and L2 = {X , Y }. Therefore, there are four layers:(A, X), (A, Y ), (B, X), and (B, Y ). The right panel shows the representation of the conventional graphmodel with multiple labels of nodes

interactions between nodes of any two different layer, given by

C = {Eαβ ⊆ Vα × Vβ;α, β ∈ 1, . . . , L, α �= β}, (4)

The two models differ in the definition of “aspect”. Kivelä’s model takes intoaccount real-life situations, e.g., a social network contains relations among varioustypes, timeline or situations.Each relation set is regarded as an aspect, namely a classifi-cation of layers, providing a comprehensive perspective. Thus, it is more considerate.The structure with aspects concept is much richer than that of ordinary networks.Possible aspects include different types of interactions or communication channels,different subsystems, different spatial locations, different points in time, and so on (DeDomenico et al. 2016). However, Boccaletti’s model is a general form that easy tounderstand. Specifically, the supra-adjacency matrix is a distinct tool for representa-tion of a multilayer network, defined as

M =

⎡⎢⎢⎢⎣

A1 I12 · · · I1LI21 A2 · · · I2L...

.... . . · · ·

IL1 IL2 · · · AL

⎤⎥⎥⎥⎦ ∈ RN×N , (5)

where A1, A2, . . . , AL are the adjacency matrix of layer 1, 2, . . . , L , respectively. Nis the total number of nodes, which can be calculated by N = ∑

1≤l≤L |V l |. Thenon-diagonal block Iαβ represents the inter-layer edges of layer α and layer β. Thus,

123

12 X. Huang et al.

Fig. 7 Supra-adjacency representation of 9/11 terrorists’ network. The supra-adjacency matrix is repre-sented as a block matrix, where the rows and columns depict the terrorists. The diagonal blocks indicatethe interactions, while the non-diagonal blocks represents the that terrorists are simultaneously active ondifferent respects of observed social actions

the interlayer edges can be represented as

I =L⋃

α,β=1,α �=β

Iαβ. (6)

Take the above-mentioned 9/11 terrorists’ network for instance, the supra-adjacencymatrix is represented in Fig. 7.

We also list some specific networks that can be represented by a multilayer networkmanner in the following.

Multiplex network is a special case of multilayer networks (Solá et al. 2013), whereall layers share the same set of nodes but may have multiple types of interactions.Some works also use multi-relational networks, multidimensional networks (Berlin-gerio et al. 2011b) or edge-colored networks (no interlayer edges) for substitution.The underlying limitations exist in the network is node-aligned, i.e. each layer hasthe same nodes but merely differs in edges. Multiplex network is a special form ofmultilayer network, where the number of nodes in each layer is consistent, and thenodes are one-to-one correspondence. This network model simplifies the complexityof the general multilayer network form and is therefore widely used to deal with somespecial problems (Kanawati 2015).

Temporal networks (or time-varying networks) differ with conventional dynamicnetworks, which focus more on the ordinal variations of connections (Kostakos 2009).

123


Fig. 8 Themonolayer network representation of the relationships of characters in “Game of Thrones” of thefirst five seasons, which contains 796 characters and 2823 links among them. The size of the node depictsthe degree centrality, e.g., Tyrion Lannister, Jon Snow, and Daenerys Targaryen, as three key roles in thestory, have larger degrees

Fig. 9 The multilayer network representation of the relationships of characters in “Game of Thrones” ofthe first five seasons. Each layer represents a season and the links between the ordinal layers represent thecorresponding relationship of characters across different seasons

The temporal network has its own set of researchmodels (Kempe et al. 2002; Kostakos2009; Tang et al. 2012a), which can illustrate the dynamic characteristics of temporalnetworks. But it must bementioned that the power of amultilayer network is that it canbe compatiblewith the representation of the temporal network and can also describe thedynamic characteristics likewise. For example, the multilayer network model we haveintroduced (Boccaletti et al. 2014) can regard the network structure at each momentas a layer, and the arrangement between different layers is in chronological order. Wecollected the relationship of characters in the Game of Thrones (the first five seasons),as shown in Figs. 8 and 9.

The study of k-partite networks starts from the complete k-partite graph (i.e., aset of graph vertices decomposed into k disjoint sets such that no two graph verticeswithin the same set are adjacent) (Brouwer and Haemers 2012). Thus, the k-partitenetwork is also described as node-colored networks (Kivelä et al. 2014), where thenodes are unacquainted in the same layer but have the other layers’ common neighborswith other nodes in the same layer. A sample of k-partite networks is shown in Fig. 10.

123

14 X. Huang et al.

Fig. 10 Illustration of a k-partite network, where k = 3. There are no intralayer edges in any layer of thenetwork, while the edges are all inter-connected across adjacent layers

Fig. 11 IllustrationofSouthwomenactivities network,which is a typical bipartite network.Thequalificationof the binary network is to check if there are links between nodes in the same type.All the links are connectingnodes with different types of nodes, corresponding with the non-diagonal elements as shown in the middlepanel. The right panel presents the corresponding supra-adjacency matrix

However, a special case of k-partite (i.e., k = 2) networks is the bipartite network (ortwo-mode network), which is more accessible by us. For example, a customer’s trans-action records of products can be represented by a “customer-product” network. Fig. 11presents a classic bipartite network, namely, South women activities network (Daviset al. 2009).

2.2 Features

Many basic metrics such as centrality, node similarity, are commonly used by commu-nity detection algorithms in monolayer networks. While in multilayer networks, themetrics need to be reformulated and adapted. Thus, in this section, some most impor-tant features of multilayer networks are introduced. Studies of structural propertiesinclude descriptors to identify the most central nodes according to various notions ofimportance (Battiston et al. 2014; De Domenico et al. 2015c, 2013; Halu et al. 2013;Solá et al. 2013) and quantify triadic relations such as clustering and transitivity (Bat-tiston et al. 2014; Cozzo et al. 2015; De Domenico et al. 2013).

123


2.2.1 Centrality

The rankingof nodes inmultilayer networks is oneof themost pressing and challengingtasks that research on complex networks is currently facing (Lü et al. 2016). Manycentrality measures have been used for single-layer networks to rank the importance ofthe nodes, such as degree centrality (Bonacich 1972), betweenness centrality (Freeman1977), closeness centrality (Freeman 1978), k-shell centrality (Carmi et al. 2007),eigenvector centrality (Bonacich 1987), PageRank (Brin and Page 1998), Leader-Rank (Lü et al. 2011), Local Centrality (Chen et al. 2012), Bridge-Rank (Salavatiet al. 2018), and so on. The above measures are widely used in the monolayer networkmodel, while the study on nodes centrality in multilayer networks is still an open issue.

Before introducing the centrality in networks, we will first cover the neighborhoodin a multiplex network. There are two definitions for the neighborhood in a multiplexnetwork: One is given by a restrict aggregation concept that j is a neighbor of i ifj is connected to i in each layer. The other is loosely defined as j is connected to iin at least one layer (Kazienko et al. 2010). Alhajj and Rokne (2014) give a trade-off definition of the above two methods: j is a neighbor of i if it is connected to iin at least m layers, where 1 ≤ m ≤ L , and L is the total number of layers. Thisdefinition may be capable of analyzing the node’s centrality in a multiplex networkwith numerous layers, but it is not ideal enough for introducing another thresholdon the number of layers for consideration. However, when applying the concept in ageneral form of multilayer networks, there are also some problems to be solved. Thefirst problem is the nodes’ relationships in a multilayer network, i.e., there may be nocorresponding nodes of i in other layers, e.g. in a k-partite network where the nodes ofeach layer are totally different. The second problem is about the presentation format ofthe node’s neighborhood, the degree centrality can hardly distinguish a node with thesame amount of interlayer edges and intralayer edges. The third problem is whetherto give equal consideration of weights on interlayer edges and intralayer edges.

In monolayer networks, one of the main centrality measures is the degree of eachnode: the more links a node has, the more important the node is. The centrality ofnode i , e.g. the degree of a node i in a multiplex network is the vector (Battiston et al.2014), defined as

ki = (k1i , k2i , . . . k

Li ), (7)

where kLi is the degree of node i in the Lth layer. We can also convert it into anotherform for simplification, given by

ki =L∑

l=1

kli . (8)

Another solution for the centrality of multilayer nodes is through the diffusionacross multiple layers. Examples of this measure include influential nodes identifica-tion, viral marketing, information diffusion, and so on (Lü et al. 2016). The networkstructure is more complicated with higher computational complexity. Some scholars

123

16 X. Huang et al.

believe that the interlayers edges should be under consideration discriminatively whencalculating the node centrality, while others insist on the viewpoint that the interlayeredges make the same contribution to the centrality calculation. Likewise, the degreecentrality measures used in multilayer networks can be substituted by other measures,e.g. Eigenvector centrality (Solá et al. 2013).

Above all, in a general form of multilayer network, the nodes may differ in eachlayer and we cannot obtain the corresponding nodes and calculate the centrality of thenodes directly. This problem has drawnmuch attention in recent years, including somepopular endeavors such as user identification across multiple networks (Carmagnolaand Cena 2009; Feng et al. 2017; Liang et al. 2015; Yang et al. 2018), social networkcoalescence, network alignment (Bayati et al. 2013), and so on.

2.2.2 Correlations

Multilayer networks encode significantly more information than their isolated singlelayers, since they incorporate correlations between the nodes in different layers andbetween the statistical properties of layers. The correlations of multilayer networksinclude interlayer degree correlations (Nicosia and Latora 2015), layers overlap-ping (Kao and Porter 2018), and so on. Some scholars point out that the communitiesin multilayer networks should consider for overlapping features, while allowing thecommunities to affect each layer in a different way, including arbitrary mixtures ofassortative, disassortative, or directed structure (De Bacco et al. 2017). A definitionof local overlap (Cellai et al. 2016) is defined as

oαβi =

∑j

θ(ωαi j )θ(ω

βi j ), (9)

where oαβi is the number of overlapping edges that are incident to node i in both layer

α and layer β,ωαi j andω

βi j are the weights of intralayer edges (i, j) in layer α and layer

β, respectively. θ(x) = 1 if x > 0 and θ(x) = 0 otherwise. Based on this concept, Kaoet al. propose a method that grouping structurally similar layers in multiplex networksand find meaningful groups of layers (Kao and Porter 2018). Considering the degreein undirected multiplex networks, the connection similarity is defined as

φαβ = 1

N

∑i

φαβi ∈ [0, 1], (10)

where φαβi is local similarity, defined as

φαβi = oαβ

i

kαi + kα

i − oαβi

. (11)

The correlations of nodes are of great importance in analyzing multilayer networkstructures. Zhan et al. improved the community detection algorithms inmulti-relational

123


social networks by utilizing triangles and latent factor cosine similarity prior meth-ods (Zhan et al. 2018). The local topological correlations between any pair of nodesfrom different layers are also utilized to calculate the centrality of nodes in multilayernetworks (Kuncheva and Montana 2015).

3 Methods

Communities are known as groups of nodes that are more strongly or frequentlyconnected among themselves than with the remainders (Aldecoa and Marín 2013).Although connecting patterns with other members are possible, they usually havehigher linking probability within the group (Fortunato and Hric 2016). Communitydetection is a fundamental issue in network science, and most existing approacheshave been developed for monolayer networks. However, many complex systems arecomposed of coupled networks through different layers, where each layer representsone of many possible types of interactions.

3.1 Problem statement

Communities in single-layer networks comprise a group of well-connected nodes,while in multilayer networks, communities reveal the relationships among nodes invarious layers. The comparison of communities on toy examples in single-layer net-works and multilayer networks is given in Fig. 12.

The problem of community detection algorithms in multilayer networks is to dividethe network into a set of disjoint cohesivemodulesC1,C2, . . . ,Ck where eachmoduleCk is comprised of a group of nodes densely connected inside and loosely connectedoutside the community. It can be described as

∪ki=1 Ci =

L∑α=1

Vα, (12)

with ∩ki=1Ci = φ for non-overlapping community detection and ∩k

i=1Ci �= φ foroverlapping scenarios. Dalibard (Dalibard 2012) gives the three requirements aboutcommunity detection works as followings:

(1) The community detection should allow for overlapping communities.(2) The detected results should be statistically significant, which means applying the

algorithm on a random null model should return no communities.(3) The detected results should be hierarchical.

Specifically, most of the existing approaches can scarcely satisfy the requirementsand hold reasonable efficiency. Besides, the evaluation metrics are also varied fromone to another, as introduced in the following subsection.

123

18 X. Huang et al.

Fig. 12 Comparison of communities in monolayer and multilayer networks. The nodes with different graylevels represent different communities. (a) Communities in a monolayer network. (b) Communities in atwo-layered network, each community shares nodes in Layer 1 and Layer 2

3.2 Evaluation functions

Quality evaluation for partition results is a complex task due to the lack of a sharedand universally accepted definition of community structures. A wide variety of qualityfunctions have been proposed to solve the community detection problem from differ-ent perspectives. Several representative metrics are analyzed and classified into threecategories (Cazabet et al. 2015):

(1) Single score metrics(2) Evaluation on generated networks(3) Evaluation on real networks with ground truth

Single score metrics employ quality functions associating a score to the communitydetection result. For example, the number of edges between partitions can be utilizedto evaluate the performance of a given algorithm. Popular metrics include Modular-ity (Newman and Girvan 2004), Modularity density (Li et al. 2008), surprise (Aldecoaand Marín 2011), and so on. These metrics are universal but often criticized withno consensus of the several meaningful levels of partitions. The comparison withgenerated networks, e.g., LFR benchmark (Lancichinetti et al. 2008) is widely usedto compare the partitions with the community affiliations. It is easy to recognize agood community and capable of evaluating variations in usual communities. How-

123


ever, sometimes the generated networks are not realistic and differ from the partitionswe want. The comparison of real networks with ground truth seems to be a reasonablesolution. Normalized Mutual Information (Danon et al. 2005), Precision, Recall, andF1-score (Herlocker et al. 2004; Perry et al. 1955) are representative methods. However,these methods depend on the priori partition labels, which are probably unknown inmost of real-world datasets.

3.2.1 Modularity, modularity density, performance and surprise

In 2004, modularity (Newman and Girvan 2004) was first proposed by Newman as anevaluation for community partitions, defined as

Q = 1

2m

∑i j

(Ai j − ki k j

2m

)δ(Ci ,C j ), (13)

wherem is the number of edges, Ai j is the element of the adjacency matrix, δ(C1,C2)

equals 1 if i and j are in the same community, otherwise 0. In 2010, Mucha et al.proposed QM (Mucha et al. 2010) for evaluating community in time-dependent, mul-tiscale and multiplex networks, defined as

QM = 1

2μ

{(Ai jα − γ

kiαk jα2mα

)δαβ + δi j C jαβ

}δ(giα, g jβ), (14)

whereμ denotes the number of links inmultiplex networks, γ is the resolution parame-ter. Ai jα represents the adjacencymatrix of nodes in layerα.C jαβ represents interlayeredge connecting node j among layer α and layer β. This metric introduces a couplingbetween communities in neighboring layers by allowing interlayer edges, while differ-ent γ enables the detection of different scale communities. However, the range of γ isrequired to be manually determined, which may be unable to obtain reasonable resultswithout an appropriate γ . Afterward, a variational version of QM is given (Pramaniket al. 2017) as

Q = 1

2m

∑i j

(Ai j − Pi j )δ(ψi , ψ j ), (15)

where δ(ψi , ψ j ) is the Kronecker delta function, it equals 1 iff ψi = ψ j , i.e. i and jbelong to the same community and 0 otherwise. The penalty term Pi j is the expectedprobability of existing an edge between nodes i and j if edges are placed at randomas

Pi j =

⎧⎪⎨⎪⎩

P1i j , if i ∈ V 1& j ∈ V 1

P2i j , if i ∈ V 1& j ∈ V 2

P12i j , if i ∈ V 1& j ∈ V 2 or i ∈ V 2& j ∈ V 1

, (16)

123

20 X. Huang et al.

where P1i j can be calculated by P1

i j = (hi ×h j )/(2|E1|) and P2i j = (hi ×h j )/(2|E2|)

and P12i j = (ci × c j )/(2|I12|). hi and h j are the intralayer degrees of nodes i and j ,

and ci and c j are the respective coupling degrees of i and j . |I12| depicts the amount ofall the interlayer edges among layers L1 and L2. Likewise, the multiplex modularityis also criticized by resolution limits (Vaiana and Muldoon 2018).

Modularity density (Li et al. 2008) was proposed to solve the resolution limitsproblem, defined as

D =c∑

i=1

L (Vi , Vi ) − L(Vi , Vi

)|Vi | , (17)

where c is the total number of communities, |Vi | is the number of nodes of the i-th community. L (Vi , Vi ) = Σ j∈Vi ,k∈Vi A jk denotes the number of edges among i-th community and L

(Vi , Vi

) = Σ j∈Vi ,k∈Vi A jk denotes the number of connectionsbetween the i-th community and other communities.

In 2010, Performance (Fortunato 2010) was proposed as a community quality func-tion, which mainly considered the number of correctly “interpreted” pairs of nodes(i.e., the nodes belonging to the same community and connected by an edge, and nodesbelonging to different communities and not connected by an edge), defined as

P =∣∣{(i, j) ∈ E,Ci = C j

}∣∣ + ∣∣{(i, j) /∈ E,Ci �= C j}∣∣

n(n − 1)/2, (18)

where E denotes the edges set, Ci and C j denote the i-th and j-th communities,respectively, n is the total number of nodes. Specifically, Coverage (Fortunato 2010)was defined as the ratio of the number of intra-community edges by the total numberof edges, given as

C =∣∣{(i, j) ∈ E,Ci = C j

}∣∣n(n − 1)/2

. (19)

In 2011, Aldecoa andMarín (2011) proposed the “Surprise” as ameasure for detect-ing communities. Different from merely considering the number of edges required ina partition, this metric takes the number of nodes into account, and it is capable ofresolving the resolution limit problem and detecting small communities (Fortunatoand Barthelemy 2007; Lancichinetti and Fortunato 2011). It is an interesting functionto measure how impossible is a given partition compared to a null model, defined as

S = − logmin(M,n)∑

j=p

(Mj

)(F−Mn− j

)(Fn

) , (20)

where F is the maximum possible number of links in the network (i.e. k[k − 1]/2,being k the number of nodes), n is the observed number of links, M is the maximumpossible number of intracommunity links observed in a partition. The parameter S,

123


which stands for Surprise, indeed measures the “surprise” (improbability) of findingby chance a partition with the observed enrichment of intracommunity links to arandom graph. The authors declare that surprise implicitly assumes a more complexdefinition of community: a precise number of units for which it is found a density oflink which is statistically unexpected given the features of the network, and in 2013,they also designed surprise maximization methods for detecting community structurein complex networks (Aldecoa and Marín 2013). As an effective metric for evaluatingcommunity structures, experiments on the human brain network (Fox and Lancaster2002; Laird et al. 2005) have also proved its priority tomodularity (Nicolini andBifone2016).

3.2.2 MDL, Pareto frontier and redundancy

The fundamental idea behind the MDL principle is that any regularity in a givenset of data can be used to compress the data, i.e. to describe it using fewer symbolsthan needed to describe the data literally (Grunwald 2004). Rosvall et al. convert thecommunity detection task into solving a coding problem following the MDL princi-ple (Rosvall and Bergstrom 2008). Analogously, the objective function of communitydetection can be considered as a multi-objective optimization problem. The optimalpartitioning for a multilayer network is achieved by maximizing a local evaluationindicator (e.g. local modularity (Chen et al. 2018b)) in each layer, i.e.,

C = argmaxC

[ f1(C), f2(C), . . . , fk(C)], (21)

where k denotes the number of communities. However, calculating an exact Paretofront is, in general, a challenging task. The most popular approximate methods aregenetic algorithms, which employ biologically inspired heuristics to attempt to trans-form randomly selected seed cases into solutions on the Pareto front using propagationMulti-objective Management (Caramia and Dell’Olmo 2008).

Berlingerio et al. define the redundancy index ρc (Berlingerio et al. 2011a), whichcaptures the phenomenon that a set of nodes constitute a community in a dimensiontend to constitute communities also in other dimensions. The redundancy figures outthe fraction of redundant links in a multi-dimensional network, defined as

ρc =∑

(u,v)∈Pc

|{α : ∃(u, v)α ∈ E}|L × |Pc| , (22)

where c represents a community, α is a layer of all the layers L = {1, 2, . . . , L}, P isa set of node pairs (u, v) existed at least one layer in a multilayer network; P is the setof node pairs existed at least two layers. Pc is the subset of P appearing in c; Pc ⊆ Pis the subset of P only containing node pairs in c. The more layers connect each pairof nodes within a community, the higher the redundancy will be.

123

22 X. Huang et al.

3.2.3 Purity, NMI, and ARI

The clustering accuracy measures are widely utilized to evaluate and compare theperformance of community detection algorithms on real-world networks with givenground-truth communities. Suppose the computed clusters Ω = {ω1, ω2, . . . , ωk}with respect to the ground truth classes C = c1, c2, . . . , ck . Purity (Zhao and Karypis2004) represents the percentage of the total number of nodes classified correctly,defined as

Purity(Ω,C) = 1

N

∑k

maxj

∣∣ωk ∩ c j∣∣ , (23)

where N is the total number of nodes, and |ωk ∩ c j | depicts the number of nodes inthe intersection of ωk and c j . To compromise the quality of the clustering against thenumber of clusters, we can utilize normalized mutual information (i.e., NMI) (Danonet al. 2005). The confusion matrix is comprised of ground communities and generatedpartitions, thereby NMI is defined as

NMI (A, B) =−2

∑CAi=1

∑CBj=1 Ni j log

Ni j NNi N j∑CA

i=1 Ni logCiN + ∑CB

j=1 N j logC jN

, (24)

where A and B denote the ground-truth communities and the detected partitions. CA

and CB are the number of groups in partition A and B, respectively. Ni j depicts theelements of the confusion matrix. Ni is the sum of the elements in row i , N j is thesum of elements in column j . N is the number of nodes. The range of NMI is [0, 1].If A = B, NMI (A, B) = 1. If A and B are completely different, NMI (A, B) = 0.Suppose an approximate size z as the number of community sets, the computationof NMI requires O(z2) comparisons, which is incapable of evaluating partitions forlarge-scale networks. In order to cope with the high computational complexity ofsuch method in recent years, several approaches (Cazabet et al. 2015; Rossetti et al.2016a, b), e.g., the precision, recall, and F1-score are employed, defined as

Precision = TP

TP + FP, (25)

Recall = TP

TP + FN, (26)

F1−score = 2 × (Precision × Recall)

Precision + Recall, (27)

with TP = true positive, FP = false positive, FN = false negative and TN= true negative.Besides, Rand Index is also a popular measure, which represents the percentage of TPand TN decisions assigns that are correct (i.e. accuracy), defined as

RI (Ω,C) = TP + TN

TP + FP + FN + TN. (28)

123


ARI (Schütze et al. 2008) is RI defined to be scaled in the range [0, 1]. All of themetrics in this subsection are in the range of [0, 1], and the higher the metric, the betterthe clustering quality is.

3.3 Algorithms classification

Early approaches either collapse multilayer networks into a weighted single-layernetwork (Berlingerio et al. 2011a; Tang et al. 2012b; Taylor et al. 2016b, a), or extendthe existing algorithms for each layer and then merge the partitions via consensusclustering (Tang et al. 2009; Papalexakis et al. 2013). However, these approacheshave been criticized for ignoring the connections across layers, thereby resulting inlow accuracy. Research to date has exhibited some novel algorithms for discoveringcommunities in multilayer networks directly (Ma et al. 2018; Pamfil et al. 2019).Generally speaking, the strategies are mainly classified into three categories (Tagarelliet al. 2017):

– Flattening methods– Aggregation methods– Direct methods

Flattening methods collapse the layers’ information into a single layer and thenconduct the traditional monolayer algorithms for detecting communities. This strat-egy is very common in multiplex networks, where the multiplex network is convertedinto a multi-relationship network (Rocklin and Pinar 2013) or a monolayer net-work (Kuncheva and Montana 2015).

Aggregation methods discover the communities in each layer and then merge themby a certain aggregation mechanism, which could be useful for removing redundantinformation. The aggregation process requires 2L comparisons and it’s very time-consuming for a temporal network with numerous layers. Thus, “layer communitiesgrouping” is proposed to reduce the redundant layers (Kao and Porter 2018). Dal-ibard proposed a parameter Pl

Ckfor each layer l ∈ L to describe the probability of

communities, and then aggregate the communities in each layer in terms of a correla-tion coefficient ραβ between layers α and β (Dalibard 2012). Numerical experimentson synthetic multilayer networks show that the analysis fails in aggregated networks,whereas themultilayer method can accurately identifymodules across layers that orig-inate from the same interaction process (DeDomenico et al. 2015a). Thus, aggregationis not recommended.

Direct methods aim to detect the community structures directly on the multilayernetwork by optimizing some quality-assessment criteria without flattening (Oselioet al. 2015). For example, Pramanik et al. defined a multilayer modularity index, i.e.,QM , and combined with the improved GN and Louvain algorithms, namely GN-QM

and Louvain-QM , respectively (Pramanik et al. 2017).There has been plenty of endeavors made in the last decades, however, research

for multilayer networks is still in infancy (Tagarelli et al. 2017), and it is promising toextract communities without collapsing multilayer networks. Several representativemethods are introduced in the following subsections.

123

24 X. Huang et al.

Table 3 Similarity measures for MNLPA algorithm

Metric in monolayer networks Modification for multilayer networks

JC(i, j) = |Γ (i)∩Γ ( j)||Γ (i)∪Γ ( j)|

∑Lα=1

(1

2ωα(|N ||U | + |N |

|U | ))

∑Lα=1 ωα

CN(i, j) = |Γ (i) ∩ Γ ( j)|∑L

α=1

(1

2ωα(|N |+|N |)

)∑L

α=1 ωα

AA(i, j) = ∑z∈|Γ (i)∩Γ ( j)| 1

log kz

∑kα=1

(1

2ωα(∑

z∈N 1log|Γ α

z | +∑

z∈N1

log|Γ αz | )

)

∑Lα=1 ωα

Γ (i) and Γ (i) denote the neighbors of node i and node j , respectively. ωα denotes the weight of layer α,N = Γ α

out_ j ∩ Γ αin_i , N = Γ α

in_ j ∩ Γ αout_i , U = Γ α

out_ j ∪ Γ αin_i and U = Γ α

out_ j ∪ Γ αin_i . Γ

αout_ j and Γ α

in_idenote out-neighbors and in-neighbors of node j and node i in layer α, respectively. Z is the intersectionof j’s out-neighbors and i’s in-neighbors. Z is the intersection of j’s in-neighbors and i’s out-neighbors

3.3.1 Improved label propagation algorithm

Label propagation algorithms utilize the propagation features of networks and havelinear complexity and reasonable results. These methods allow the nodes to adopt newcharacteristics depending on the behavior of their neighbors, e.g. adopts labels of thebiggest amount of its neighbors. The process of LPA is shown as follows:

Step 1: Traverse the network and assign a unique label to each node.Step 2: Establish a random order of the node’s revision.Step 3: Each node is revised in the assigned order and adopts the most frequent label

of its neighbors.Step 4: The process is performed iteratively until the algorithm converges and no label

changes occur anymore.

Inspired by the traditional LPAprocess, Alimadadi et al. (2019) redefined the neigh-borhood in multilayer networks and then proposed MNLPA to detect communities ina weighted and directed Facebook activity network. The algorithms are summarizedas followings:

Step 1: Each node u is initiated with a unique label, then the neighbors of u in alllayers are obtained, marked as Nu .

Step 2: Calculate the similarity (measures are listed in Table 3) of u between the nodev in Nu , and mark v when the similarity score is more than a given thresholdσ .

Step 3: Repeat the following steps until the stop criterion is satisfied:

– Nodes are ordered randomly;– For each node u, each marked similar neighbor sends out its label to u, andmark the node u with maximum labels.

Step 4: Divide communities by the nodes with the same labels.

The MNLPA is praiseworthy with efficiency, and capable of dealing with weightedand directed networks, but also criticized by instability. The partition result is sensitive

123


age city gender

Closen

ess

Spent

Tim

e

Trus

t

Closen

ess

Spent

Tim

e

Trus

t

Closen

ess

Spent

Tim

e

Trus

t

0.00

0.25

0.50

0.75

0.0

0.1

0.2

−0.1

0.0

0.1

applications

hom

ophi

ly methods

LPA

MNLPA

Fig. 13 Comparison of LPA and MNLPA algorithms conducting on Facebook datasets. The homophilyobtained by MNLPA algorithm is marked by gray bars, thus the resultant communities are similar and fitthe definition of the community to some extent

Fig. 14 Three synthetic multilayer networks for evaluating the proposed MNLPA. The top three panelsshow the multilayer networks, in which each has three layers with community labels generated by LFRbenchmark (Lancichinetti et al. 2008). The bottom three panels show the relevant supra-adjacency matrices

to the threshold parameter in MNLPA and the density of the network dataset. As theydeclared, the MNLPA algorithm is verified by experiments on real-world datasets andthe results are reprinted in Fig. 13.

As the Facebook datasets are not provided in their works, we construct severalsynthetic three-layered networks, as plotted in Fig. 14.

Experiments on the constructed synthetic networks suggest that MNLPA algorithmis fastidious about the parameters. The changing ofmodularitywith the varying thresh-old δ is plotted, as shown in Fig. 15.

As shown in Fig. 15, the performance of the proposed MNLPA is not satisfied.With an increasing threshold in [0.2, 0.4], the modularity increased, thus we guess thethreshold should be greater than 0.4 approximately. The Facebook datasets utilized in

123

26 X. Huang et al.

Fig. 15 The modularitiestendency obtained by theMNLPA algorithm conductingon three different syntheticnetworks changes with thevarying threshold δ

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.25 0.50 0.75

δ

mod

ular

ity

nodes

60 nodes

120 nodes

180 nodes

the experiments are weighted and directed, which may make prominent differencesin the experimental results. In brief, the MNLPA is suitable for large-scale networksunder certain conditions (e.g., weighted and directed), while for general multilayernetworks, it’s necessary to make modifications to improve the performance.

3.3.2 Nonnegative matrix factorization methods

Nonnegative matrix factorization (NMF) was proposed by Lee and Seung (2001).It aims to factorize the original nonnegative matrix into the product of two othernonnegative matrices. For applications in community detection methods, the originalnonnegative matrix can be an adjacency matrix, thereby the objective (loss) functioncan be presented as

minU≥0,V≥0

L(A,UV T ) =∥∥∥A −UV T

∥∥∥2F

, (29)

where A is a n × n adjacency matrix, and both U and V are n × k matrices. The rankk corresponds to the number of divided communities. It has been widely utilized indetecting communities in complex networks (Jiao et al. 2017; Liu et al. 2017;Wu et al.2018).

Recently, Ma et al. applied this method (S2j-NMF) to community detection formultilayer networks (Ma et al. 2018). They propose a quantitative function (i.e. mul-tilayer network modularity density) and prove the trace optimization of multilayermodularity density is equative to the objective functions of the community detectionalgorithms (e.g. k-means (MacQueen 1967), NMF, spectral clustering (Ng et al. 2002),multiview clustering for multilayer networks, etc.). The modularity density QD for{Vc}kc=1 is defined as

QD

({Vc}kc=1

)=

k∑c=1

L (Vc, Vc) − L(Vc, Vc

)|Vc| , (30)

where QD({Vc}kc=1) is themodularity density of community partitions, k is the numberof partitions, Vc depicts the partitions after removing Vc. L(Vi , Vj ) calculates theconnections between Vi and Vj , defined as

123


L(Vi , Vj ) =∑

p∈Vi ,q∈Vj

wpq , (31)

where p and q are the nodes of the partition Vi and Vj , respectively. wpq is the weightof the edge (p, q) and equals 1 in unweighted networks. The objective function istransformed into a multi-objective optimization problem as

QGD

({Vc}kc=1

)= 1

m

m∑l=1

k∑c=1

Ll (Vc, Vc) − Ll(Vc, Vc

)|Vc| , (32)

where Ll(Vi , Vj ) is the same with equation (31) applied in l layer. QGD({Vc}kc=1) is

the objective function of the partitions in multilayer network G. Thus, the optimalpartitioning {Vc}kc=1 for the multilayer network by maximizing the modularity densityin each layer can be represented as

⎧⎪⎪⎨⎪⎪⎩

max(Q1

D({Vc}kc=1))

max(Q2

D({Vc}kc=1))

· · ·max

(Qm

D({Vc}kc=1))

. (33)

Afterward, dense subgraphs are discovered by employing a greedy search strategyin multilayer networks. The conventional NMF algorithm combined with a factorizedbasis matrix and various coefficient matrices are applied to each layer. Finally, theexperiments are conducted on several datasets, which verifies the proposed method.

The complexity of this method is O(mn2k), where m is the number of layersand k is the number of partitions. Thus, it is probably not acceptable for large-scalenetworks. Besides, as the authors mentioned, the algorithm is based on multiplexnetworks, which is not capable of handling the general form of multilayer networks.The algorithm relies on prior information and the number of target communities, andthe decomposition process might be time-consuming.

3.3.3 Randomwalk methods

Kuncheva et al. propose a community detection algorithm, namely LART (LocallyAdaptive Random Transitions) for the detection of communities that are shared byeither some or all the layers in multiplex networks (Kuncheva and Montana 2015).They employ the supra-adjacency matrix M and define the transition probabilities offour possible moves among the nodes, described as

123

28 X. Huang et al.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

P(i, j)(i,k) = M(i, k)(i, k)

ki,k

P(i, j)( j,k) = M(i, k)( j, k)

ki,k

P(i,k)(i,l) = M(i, k)(i, l)

ki,kP(i, j)( j,l) = 0

, (34)

where ki,k is the multiplex degree of node vki in M defined as ki,k = ∑j,l M(i,k)( j,l),

P(i, j)( j,k) depicts the transition probability from node i of layer k (i.e., vki ) to node jof the layer l (i.e., vlj ). The probability to move from node vki to node vlj is zero wheni �= j and k �= l since there cannot exist a direct move where there is no connection.The transition probabilities are represented as a matrix P of the random walk processand written as

P = D−1M, (35)

where D is the diagonal matrix defined by the multiple node degrees. A dissimilaritymatrix S(t) which depends on the multiplex random walk of steps t is defined. Thedissimilarity matrix is defined according to the nodes i and j are in the same layer ordifferent layers, denoted by

S(t)(i,k)( j,k) =

√√√√√N∑

h=1

L∑m=1

(P t

(i,k)(h,m) − P t( j,k)(h,m)

)2

k(h,m), (36)

S(t)(i,k)( j,l) = √s1 + s2 + s3, (37)

where s1, s2, s3 are defined as

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

s1 = ∑Nh=1

(P t

(i,k)(h,k)√k(h,k)

− P t( j,l)(h,l)√k(h,l)

)2

s2 = ∑Nh=1

(P t

(i,k)(h,l)√k(h,l)

− P t( j,l)(h,k)√k(h,k)

)2

s3 = ∑Nh=1

∑Lm=1;m �=k,l

(P t

(i,k)(h,m)−P t

( j,k)(h,m)

)2

k(h,m)

. (38)

Afterward, the agglomerative clustering is utilized to merge nodes in communities.The multiplex modularity QM is employed to evaluate the quality of partitions. Theprocess of LART is shown as follows:

Step 1: Assign each node in each layer to its own community.Step 2: Merge nodes based on the average linkage criterion using the distance matrix

S and obey the principle of the merged community has at least one within-layer or interlayer connection.

123


Fig. 16 The comparison of theproposed LART algorithm withMM and PMMalgorithms (Kuncheva andMontana 2015) on the fivesimulated scenarios of thesynthetic network

0.00

0.25

0.50

0.75

1.00

S1 S2 S3 S4 S5scenarios

NM

I

algorithms

LART

MM

PMM

Step 3: Merge the nodes only if the maximum QM is reached.Step 4: Obtain the shared and non-shared communities.

The experiments are conducted on synthetic multiplex networks, and the experi-mental results are shown in Fig. 16.

The proposed LART algorithm is conducting on five different scenarios and theexperimental result demonstrates the performance of the proposed algorithm. How-ever, the algorithm is limited to multiplex networks, and the real-world networks aremuch more complicated. Hence, the performance of the LART algorithm for real-world datasets is uncertain.

3.3.4 Multi-objective optimization methods

Pizzuti and Socievole (2017) proposed the Multi-layer many-objective Optimizationalgorithm (MLMaOP), in which they formulated the community detection problem inmultilayer networks as a many-objective optimization problem and a given objectiveis contemporarily optimized on all the network layers. In their work, they give themulti-objective optimization problem (MOP) as

minx

F(x) = ( f1(x), f2(x), . . . , fd(x)) subject to x ∈ X , (39)

where d is the number of objective functions, x = (x1, x2, . . . , xn) ∈ X is the decisionvector with a domain of definition X ⊆ Rn, F : X → Z is the mapping from thedecision space X to the objective space Z . When d ≥ 3, an MOP is referred to asMany Objective Optimization Problem (MaOP) (Farina and Amato 2002). Pareto-dominance relation is used to define a partial ordering in the objective space. Thus,the problem of community detection in multilayer networks using MaOP is definedas

min F(P) = (F1(P), F2(P), . . . , Fd(P)) subject to P ∈ Ω, (40)

where each Fα : Ω → R computes the value of the objective function only on thelayerGα . For the main purpose is to get a maximized Q, so Fα(P) = −Qα(P)meansthat the greater Q, the smaller Fα(P) partitions on each layer. The main process ofMLMaOP is shown as follows:

Step 1: Initialize a rand partition by using the adjacency matrix of projected M .

123

30 X. Huang et al.

0.00

0.25

0.50

0.75

1.00

P1 P2 P3Partitions

NM

I

algorithms

CLECC−BD

Generalized Canonical Correlations

MLMaOP−cspa

MLMaOP−mf

MLMaOP−proj

Projection Average

Projection Binary

Projection Cluster−based Similarity Partition

Projection Neighbors

Fig. 17 The comparison of MLMaOP algorithm with competitors on SSRM dataset (Loe and Jensen 2015)

Step 2: Traverse the partition in all the layers, evaluate the objection function on Gα

to obtain Fα(P). Assign a rank based on Pareto dominance and then combineparents and offspring partition into fronts.

Step 3: Select the best points, and apply the variation operators and create the nextpartition.

Step 4: Choose a solution from the Pareto front.

The comparison of MLMaOP algorithm with other approaches (Loe and Jensen2015) is shown in Fig. 17.

The proposed algorithm with three different strategies is competitive on the parti-tions P1, while on P2 and P3, we can see that the NMI results obtained by the otheralgorithms are better. Besides, theMLMaOP algorithm suffers from a low convergencerate to Pareto front and is likely to be time-consuming for detecting communities inlarge-scale networks.

3.4 Discussion

In the last decade, a plethora of approaches have been proposed to address the com-munity detection problem with enormous network data. We list several representativemethods (from 2009 to 2019), as shown in Table 4.

Table 4 shows that most of the presented methods are holding relatively high com-plexity, where GN-QM , Louvain-QM and LART methods are based on multiplexmodularity maximization and unfavorable on general multilayer networks. Moreover,themajority are designed formultiplex networks, which require the nodes in each layershould be aligned. As we have introduced in the previous subsection, some improvedversion of classic monolayer algorithms, e.g., GenLouvain has been regarded as abenchmark and is really worth expecting for general multilayer networks. We cananticipate four prospective directions, i.e., randomwalk-based method, tensor decom-position, nonnegative matrix factorization, and modularity optimization will receiveincreasing attention over time.

In addition to the above-mentioned directions, quite a part of algorithms focuson overlapping community detection (Liu et al. 2018) and local community detec-tion (Interdonato et al. 2017; Jeub et al. 2015; Li et al. 2019). On the one hand,

123


Table4

Abriefcomparisonof

commun

itydetectionmetho

dsin

multilayer

networks

Nam

eClassificatio

nStrategy

Com

plexity

aNetwork

PMM

(Tangetal.2

009)

Aggregatio

nMultilayer

modularity

maxim

ization

O(n

3L)

Multi-dimensional

MULT

ICLUS(Papalexakisetal.2

013)

Agg

regatio

nMinim

umdescriptionleng

thO

(mnIC

2)

Multip

lex,

bipartite

GRAPH

FUSE

(Papalexakisetal.2

013)

Aggregatio

nTensor

analysis

O(n

3L)

Multip

lex

ABACUS(Berlin

gerioetal.2

013)

bAggregatio

nMultilayer

modularity

maxim

ization

–Multi-dimensional

DNMF(Jiaoetal.2

017)

Aggregatio

nNonnegativ


O(n

2L)

Multip

lex,

tempo

ral

EMCD(Tagarellietal.2

017)

cAggregatio

nModularity

maxim

ization

O(I

(m+

LC

))Multip

lex

Multilink(M

ondragon

etal.2

018)

Aggregatio

nMultilinksimilarity

O(m

2)

Multip

lex

M-EMCD*(M

andaglio

etal.2

018)

Agg

regatio

nMod

ularity

maxim

ization

O(I

(m+

LC

))Multip

lex

M-M

otif(H

uang

etal.2

019)

Agg

regatio

nMerge

partition

sacross

layers

O(n

log(n)L2)

Multilayer

MEMM

(Zhang

etal.2

019)

Agg

regatio

nMultilayer

edge

mixture

mod

elO

(n2)

Multip

lex

HSB

M(Paezetal.2

019)

Agg

regatio

nHierarchicalS

BM

andBayes

O(n

2LC2)

Multip

lex

Variatio

nal-Bayes

(Alietal.2

019)

Agg

regatio

nSB

MandvariationalB

ayes

O(n

2L2C

)Multip

lex,

weighted

GenLouvain

(Jutlaetal.2

011)

Direct

Multip

lexmap

equatio

nO

(n2logn)

Multip

lex,

tempo

ral

MultiG

A(A

melio

andPizzuti2

014b

)Direct

Geneticrepresentatio

nO

(In2

L)

Multip

lex

MultiM

OGA(A

melio

andPizzuti2

014a)

Direct

Multi-objectiveoptim

ization

O(L

Cn2

)Multip

lex

CLAN(D

abideenetal.2

014)

Direct

Variatio

nallabelpropagation

O(L

InK

)Multip

lex,

tempo

ral

LART(K

unchevaandMon

tana

2015

)Direct

Rando

mwalk

O(m

3)

Multip

lex

Multip

lex-Infomap

(DeDom

enicoetal.2

015a)

Direct

Multip

lexmap

equatio

nO

(n2)

Multip

lex

LocalNCPs

(Jeubetal.201

5)d

Direct

Localrandom

walk

≥O

(nIK

L)

Multip

lex

BAZZI(Bazzietal.2

016)

Direct

Multilayer

modularity

maxim

ization

O(nI2

L)

Multip

lex,

tempo

ral

ML-LCD(Interdo

nato

etal.2

017)

Direct

Localobjectivefunctio

nmaxim

ization

≥O

(C3K2L)

Multip

lex

123

32 X. Huang et al.

Table4

continued

Nam

eClassificatio

nStrategy

Com

plexity

aNetwork

GN-Q

M(Pramanik

etal.2

017)

Direct

Maxim

umbetweennessedgesremoval

O(nm2)

Multip

lex

Lou

vain-Q

M(Pramanik

etal.2

017)

Direct

Modularity

optim

ization

O(m

3)

Multip

lex,

weighted

MLMaO

P(PizzutiandSo

cievole20

17)e

Direct

Multi-objectiveOptim

ization

–Multip

lex

NFC

(Aslak

etal.2

018)

Direct

Rando

mwalkandInfomap

O(m

(n+

m))

Multip

lex,

tempo

ral

S2-jNMF(M

aetal.2

018)

Direct

Nonnegativ


O(rn2

km)

Multip

lex

MNLPA

(Alim

adadietal.20

19)

Direct

LabelPropagation

O(nk)

Multip

lex,

directed,w

eighted

IterModMax

(Pam

filetal.2

019)

Direct

SBM

andModularity

maxminization

O(n

2L)

Multip

lex,

tempo

ral

TMSC

D(K

unchevaandMon

tana

2017

,20

19)

Direct

Spectralgraphwavelet

O(n

2L)

Multip

lex,

tempo

ral

CMNC(Chenetal.2

019)

Direct

Tensor

Decom

positio

nO

(n3L)

Multip

lex

MCD-B

erlin

gerio(B

erlin

gerioetal.201

1a)f

Flattening

Employingmon

olayer

algo

rithms

–Multi-dimension

al

CDHIA

(Tangetal.2

012b

)Flattening

networkintegrationandk-means

≥O

(nIC

K)

Multi-dimensional

Aggregatio

nPan

(Pan

etal.2

018)

Flattening

Cuttin

gedgeswith

weight<

threshold

O((m

+n)L)

Multip

lex,

weighted

ParticleGao

(Gao

etal.2

019)

Flattening

ParticleCom

petition

O(nIC

L2)

Multip

lex,

directed,w

eighted

anandm

arethenu

mberof

nodesandedges,respectiv

ely.Listhenumberof

layers,K

istheaveragedegree

ofno

des,Idenotestheiteratio

ntim

es.

bThe

complexity

ofABACUSdepend

son

thecomplexity

oftheem

ployed

mon

olayer

algo

rithms,e.g.,O

(n)from

LPA

(Raghavanetal.200

7)andwith

totalcom

plexity

ofO

(LCN2).

cIdeno

testhenu

mberof

iteratio

nsto

convergenceatalocalo

ptim

um;C

isthenu

mberof

commun

ities.

dLocalNCPs

isalocalcommun

itydetectionalgo

rithm.Fo

ragivenno

de,thecomplexity

isapprox

imatelyO

(LIn

K/C

).Fo

rC

partition

s,theminim

umcomplexity

isO

(LIn

K).

eThe

complexity

ofMLMaO

Pdependson

anuncertainconvergenceprocess,therebymarkedwith

“–”.

fThisalgorith

mprovidesafram

eworkviaflatte

ning

amulti-dimension

alnetworkintoaweigh

tednetwork,andthen

employsthe

existin

gmon

olayeralgorithmsforcommun

itydetection,

therebythecomplexity

isun

certain

123


with the increasing of network scale, global computation becomes time-consuming,which promoting local community detection into our view. Liu et al. (2017) proposedan improved multi-objective evolutionary approach for community detection in multi-layer networks.Aiming at solving the local community detectionproblem, they employa string-based representative scheme and genetic operation and local search. However,the algorithm adapts the strategy of conducting the Louvain algorithm (Blondel et al.2008) on each layer and then merges the partitions, which seems to deviate from themultilayer community concept. More than that, comparisons with other competitorsare not provided. On the other hand, overlapping communities are also ubiquitousin multilayer networks (De Domenico et al. 2016), i.e., some nodes are attached tomultiple partitions simultaneously (Chen et al. 2016). Kao and Porter (2018) proposeda method based on computing pairwise similarities between layers and then executingcommunity detection for grouping structurally similar layers in multiplex networks.The algorithm is verified in both synthetic and empirical multiplex networks. As mostof the compared algorithms are designed for multiplex networks, there’s still a greatdeal of works to do in community detection in the general multilayer networks.

In brief, the research on community detection for multilayer networks is just in itsinfancy. At the time of this writing, there is still no standard algorithms for generalmultilayer networks and quite a few problems remain to be solved, such as the opti-mization of algorithm process to avoid time-consuming procedures, the extending ofalgorithms for applying in general form of multilayer networks, the simplification ofmathematical model, and so on.

4 Applications

The study of detecting community structures in multilayer networks is experiencing ablossom in the last decade. Relevant researches cover various aspects among our dailylife such as analyzing influential users in multiple social platforms (Al-Garadi et al.2018), finding organization of proteins in a biological system (Gosak et al. 2018) andmanaging urban transportation system with various traffic manners (Liu et al. 2019),etc. The following subsections summarized applications of community detection viaa multilayer network framework.

4.1 Temporal networks partition

Community detection in temporal networks, i.e., temporal community detection, isrequired to find how communities emerge, grow, combine, and decay in an evolvingprocess (Kawadia and Sreenivasan 2012). A common approach to detect temporalcommunities is to obtain communities independently in each snapshot by utilizingstatic methods and then map the partitions between two snapshots together as manyas possible. Obviously, it fails to achieve the goal of revealing the evolving processbecause such methods do not adequately use partitions found in past snapshots toinform the identification for the optimal partition on the current snapshot (Jiao et al.

123

34 X. Huang et al.

2017). Thus, as the foremost step of modeling, the traditional graphmodel is incapableof presenting inter-connections in a temporal network.

Multilayer network model is commonly employed in the study of time-varyingnetworks (or temporal networks, multi-slice networks), in which the time snapshotsare modeled as layers and the layers are ordered by a certain sequence. However,there’s a foundational question: Across how many layers must a community persistin order for layer aggregation to benefit detection? To solve this problem, a layeraggregation approach (De Domenico et al. 2014) is proposed to reduce data size or asa data filter to benefit network-analysis outcomes. SinceMucha et al. (2010) introducedthe multiplex modularity optimization method, numerous attempts were made in thisfield (Drugan et al. 2011; Nguyen et al. 2011; Li and Garcia-Luna-Aceves 2013),which opened up a upsurge in unveiling the communities in time-varying networks.Taylor et al. (2017) proposed the random matrix theory and found layer aggregationto significantly influence detectability. The detectability limitation is described as theability of network structure to form a community, i.e., if the community structureis too weak, it cannot be found upon inspection of the network (Lancichinetti andFortunato 2011). When the aggregative network corresponds to the summation ofthe adjacency matrices encoding the network layers, aggregation always improvesdetectability. The research is beneficial to understand the contraction of network layersand analyze pairwise-interaction data to obtain sparse network representations. Theapplication of layer aggregation can be used for anomaly detection in network data,e.g., in cybersecurity, detecting harmful events such as attacks, intrusions, and fraud.

4.2 Transportation networks optimization

On account of the critical role of transportation system in modern society, the studyon traffic dynamics has become one of the most successful applications of complexnetwork theroy. However, the vast majority of researches treat transportation networksas an isolated system, which is inconsistent with the fact that many complex networksare interrelated in a nontrivial way (Du et al. 2016). Analogously, the transportationsystem has a variety of traffic manners, such as bus, subway, tram, high-speed train,airline, ship, etc, hence a comprehensive study should cover many of such manners.Early researches of traffic networks mainly focus on a single traffic way and ignore theinteractions between their counterparts (Calimente 2012; Chen et al. 2014). Du et al.(2016) utilized a two-layered traffic network to study the distribution-based strategyand improved the generating rate of passengers using a particle swarm optimizationalgorithm. The multilayer network model utilized in this work is an idealized trans-portation system, in which each layer has a different topology and supports differenttraveling speeds. The passengers are allowed to travel along the path of minimal trav-eling time and with the additional cost they can transfer from one layer to another toavoid congestion. The research indicates that a degree centrality-based strategy is notoverly beneficial in enhancing the performance of the system. However, starting fromsuch a strategy and reassigning transfer costs using a particle swarm optimization algo-rithm improve the capacity and several other properties of the system at a reasonablecomputational cost. The research is rewarding to the selection of traffic manners and

123


exemplifies how multilayer network models are applied in the urban transportationsystem.

Inspired by the complex network theory and the multilayer network representation,Hong and Liang (2016) analyze the Chinese airline transportation system with themultilayer network framework, in which each layer is defined by a commercial airline(company) and theweights of links are set by the number of flights, the number of seatsand the geographical distance between pairs of airports, respectively. By calculatingthe clustering coefficient, average shortest path length, and assortativity coefficientof the airports, the research has shown that the Chinese airline is of considerablyhigher value of a maximal degree and betweenness than the other top airlines. Dinget al. (2018) proposed a method for measurements in areas of Kuala Lumpur (i.e., thenational capital of Malaysia) to detect communities. The multilayer network modelemployed in their research contains the railway layer and urban street layer, whichmainly focuses on detecting the changing structures of a rail network and mining inurban network communities. The experimental results suggest that rail network growthtriggers structural and community changes, i.e., when an upper-layer rail networkgrows from a simple tree-like network to a more intricate form, the network diameterand average shortest path length decrease dramatically. The growth of the networkallows the remainder of the network to be easily visited, which provides suggestivepatterns for city development. Yildirimoglu and Kim (2018) analyzed the urban trafficnetwork by combing bus lines, passenger trajectories, and vehicle trajectories togetherand formed a three-layered network. By applying the Louvain algorithm (Blondel et al.2008) independently on the three layers, they found that aggerated patterns can shapegeographically well-connected communities in the urban traffic network. The spatialstructure is quite alike for the bus and passenger layers, which benefits transit authorityin making location decisions. The research is beneficial from a planning perspectivethat sub-regional borders designate the influential areas around local centers, shoppingdistricts, school zones, etc., and cities can develop policies in order to improve theaccessibility to them and enhance network performance.

4.3 Social network analysis

Another hot-point of community detection research in multilayer networks is socialnetwork analysis (Alhajj and Rokne 2014). Social networks have been studied fairlyextensively over the last couple of decades, mainly in the general context of analyzinginteractions between people in order to determine important structural patterns in suchinteractions. With the utilization of plentiful data resources from online social mediasuch as Facebook, Twitter, and Flickr, there’s an increasing tendency in discoveringcommunity structures in such time-varying social networks (Alimadadi et al. 2019;Rozario et al. 2019; Zhou et al. 2007, 2016). The emergence of online social networkshas altered millions of web users’ behavior so that their interactions with each otherproduce huge amounts of data on various activities. Facebook and Twitter, as thetop-two popular social media in our daily life, have been widely employed for socialnetwork analysis in recent years (Alimadadi et al. 2019; Türker and Sulak 2018).

123

36 X. Huang et al.

The analysis of social networks is usually accompanied by various applicationssuch as information propagation, internal trades analysis, influential spreaders identi-fication, and so on. Diffusion processes, like the propagation of information or thespreading of diseases, are fundamental phenomena occurring in social networks.While the study of diffusion processes in single networks has received a great deal ofinterest from various disciplines for over a decade, diffusion on multilayer networksis still a young and promising research area, presenting many challenging researchissues (Salehi et al. 2015). Numerous attempts have been made to uncover the com-munity structures in international trades, typically represented as bipartite networksin which connections can be established between countries and industries (Alves et al.2019). Biondo et al. (2017) present amultilayer networkmodelwith contagion dynam-ics, which is able to simulate the spreading of information and the transactions phaseof a typical financial market. In their two-layered network framework, the first layercomprises the trading decisions of investors, and the second layer is constructed of theinformation dynamics, which is fruitfully beneficial to explain the aggregate behaviorof markets. Basaras et al. (2017) proposed an effective method to detect influentialspreaders in multilayer networks based on the underlying community structures. Theexperimental evaluation shows that the proposed method outperforms the major com-petitors proposed so far for either single-layer or multilayer networks.

The above-mentioned applicationsmainly focus on a local structure or some certaincontext-based community detection for the case of large volume and various dynamicchanges of networks. Thus, it is of great significance in designing some smart algo-rithms to mine the valuable information among plentiful social network resources.

4.4 Research on biological systems

Biological systems, from a cell to the human brain, are intrinsically complex (Ma’ayan2017). Multilayer networks, described by an intricate network of relationships acrossmultiple scales, are most widely employed in representing such systems. The majorityof the biological processes are constituted by a group of proteins that are connecteddensely (Cui et al. 2012). The protein-protein interaction (PPI) network contains thecommunications among the protein groups that communicate with each other closely,which can be used to predict the complexity of the function of normal proteins (Srihariet al. 2017). In general, there are two typical protein communities: protein complexesandprotein functionalmodules. Protein complexes are sets of proteins that interactwitheach other to execute a single multimolecular mechanism. Protein functional modulesare sets of proteins that participate in a particular biological process, and interact witheach other at different time and places (Spirin andMirny 2003). Recently, several stud-ies are highlighting how simple networks, i.e., obtained by aggregating or neglectingtemporal or categorical descriptions of biological data, are not able to account for therichness of information characterizing biological systems (De Domenico 2018). Chenet al. (2018a) proposed an MLPCD algorithm by integrating Gene Expression Data(GED) and a parallel solution of MLPCD using cloud computing technology. Theyreconstructed the weighted protein-protein interaction (WPPI) network by combiningPPI network and related GED, and then defined simplified modularity as the ratio of

123


in-degrees and out-degrees of proteins in a community. By utilizing an improved Lou-vain algorithm (Blondel et al. 2008), they have achieved the goal of detecting proteincomplexes and protein function modules.

Although non-overlapping communities are more commonly studied in networkneuroscience, a model of community structure that allows for overlapping networksoffers a more realistic presentation of brain network organization (Wu et al. 2011).Taking overlapping communities into consideration, Zhang et al. (2018a) propose acentral edge selection (CES) based community detection algorithms for PPI networks.Experimental results on three benchmark networks and two PPI networks indicatethe excellent performance of the proposed CES algorithm. Kurmukov et al. (2017)propose a framework to compare both overlapping and non-overlapping communitystructures of brain networks within the machine learning settings. The performanceof the proposed framework is verified in the task of classifying Alzheimer’s disease,mild cognitive impairment, and healthy participants. Pan et al. (2018) present anaggregation approach to detect communities in multilayer biological networks, whichfirst constructs a consensus graph form multiple networks and then applies traditionalalgorithms to detect communities. Inspired by the fact of few shared edges existedamong different networks, they merge the weights of edges from different layersand cut off the nodes with low weights. The approach is simple but limited by theapplication scenarios.

Another notable direction of biological research is about human brain networks.Cantini et al. (2015) propose a multi-network-based strategy to integrate different lay-ers of genomic information and use them in a coordinatedway to identify diving cancergenes. The multi-networks they focus, combine transcription factor co-targeting,microRNA, cotargeting, protein-protein interaction, and gene co-expression networks.The combination of different layers benefits extracting from the multi-networks indi-cations on the regulatory pattern and functional role of both the already known and thenew candidate diver genes. Sanchez-Rodriguez et al. (2019) introduce an approachfor the detection of a modular organization by considering the temporal scales of theinformation flow over large-scale brain graphs, and several organizational patternsexisting in the brain anatomical and functional networks are found. The structuresmay coexist together, in a dynamical way that is given by the temporal scales of theactivity they produce, guaranteeing functional independence and coordination.

In brief, discovering the underlying patterns in biological networks is experiencing ablossom.With the development of network science, multi-biological networks provideplentiful data resources than ever before, which requires us to dedicate more to thispromising field.

5 Outlook

As interdisciplinary research with a variety of prospective applications, complex net-work has been receiving increasing attention from the scientific community. Inspiredby prosperous real-world scenarios such as social networks, biological networks, andtransportation networks, extensive researches have been dedicated to the extractionof non-trivial knowledge from such networks. Along with the further study, scholars

123

38 X. Huang et al.

come to realize many systems are inherently represented by a multilayer network,in which edges exist in multiple layers that encode differently but potentially related,types of interactions, and it is important to uncover the interlayer community structuresin a complex system.

This paper first presents the various formats of multilayer networks and thenintroduces the two basic mathematical models. Subsequently, the quality evaluationmeasures and several typical community detection algorithms are introduced, includ-ing label propagation-based algorithm, nonnegativematrix factorization, randomwalkmethods, andmulti-objective optimization methods, and so on. After a comprehensiveanalysis of the above-mentioned methods, we conclude that most of the existing meth-ods are designed formultiplex networks, i.e., the nodes in each layer are aligned, whichlimits the research on universal multilayer network format. Besides, the algorithmsare with high computational complexity and can hardly obtain reasonable partitionsamong large-scale multilayer networks. A great deal of works remain to be done in thefuture, such as designing more efficient algorithms for temporal networks with numer-ous layers and exploring the community structures in special formats of multilayernetworks.

Acknowledgements We would like to thank the anonymous reviewers for their careful reading and use-ful comments that helped us to improve the final version of this paper. This work is partially supportedby Liaoning Natural Science Foundation under Grant No. 20170540320, the Science Research Projectof Liaoning Provincial Department of Education under Grant Nos. L2015167, L2015173, the DoctoralScientific Research Foundation of Liaoning Province under Grant No. 20170520358.

Compliance with ethical standards

Conflicts of interest The authors declare that they have no conflict of interest.

OpenAccess This article is licensedunder aCreativeCommonsAttribution 4.0 InternationalLicense,whichpermits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,and indicate if changes were made. The images or other third party material in this article are includedin the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. Ifmaterial is not included in the article’s Creative Commons licence and your intended use is not permittedby statutory regulation or exceeds the permitted use, you will need to obtain permission directly from thecopyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

References

Ahmed I, Witbooi P, Christoffels A (2018) Prediction of human-bacillus anthracis protein-protein interac-tions using multi-layer neural network. Bioinformatics 34(24):4159–4164

Al-Garadi MA, Varathan KD, Ravana SD, Ahmed E, Mujtaba G, Khan MUS, Khan SU (2018) Analysisof online social network connections for identification of influential users: survey and open researchissues. ACM Comput Surv 51(1):1–37

Aldecoa R, Marín I (2011) Deciphering network community structure by surprise. PLoS ONE 6(9):e24195Aldecoa R, Marín I (2013) Surprise maximization reveals the community structure of complex networks.

Sci Rep 3(1):1–9Alhajj R, Rokne J (2014) Encyclopedia of social network analysis and mining. Springer, Berlin

123

http://creativecommons.org/licenses/by/4.0/


AliHT,Liu S,YilmazY,Couillet R,Rajapakse I,HeroA (2019)Latent heterogeneousmultilayer communitydetection. In: ICASSP 2019-2019 IEEE international conference on acoustics, speech and signalprocessing (ICASSP). IEEE, pp 8142–8146

Alimadadi F, Khadangi E, Bagheri A (2019) Community detection in facebook activity networks andpresenting a new multilayer label propagation algorithm for community detection. Int J Mod Phys B33(10):1950089

Alves LG, Mangioni G, Cingolani I, Rodrigues FA, Panzarasa P, Moreno Y (2019) The nested structuralorganization of the worldwide trade multi-layer network. Sci Rep 9(1):1–14

Amelio A, Pizzuti C (2014a) Community detection in multidimensional networks. In: 2014 IEEE 26thinternational conference on tools with artificial intelligence. IEEE, pp 352–359

Amelio A, Pizzuti C (2014b) A cooperative evolutionary approach to learn communities in multilayernetworks. In: International conference on parallel problem solving from nature. Springer, pp 222–232

Ansari A, Koenigsberg O, Stahl F (2011) Modeling multiple relationships in social networks. J Mark Res48(4):713–728

Aslak U, Rosvall M, Lehmann S (2018) Constrained information flows in temporal networks reveal inter-mittent communities. Phys Rev E 97(6):062312

Baltakiene M, Baltakys K, Cardamone D, Parisi F, Radicioni T, Torricelli M, de Jeude J, Saracco F (2018)Maximum entropy approach to link prediction in bipartite networks. arXiv preprint arXiv:1805.04307

Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512Basaras P, Iosifidis G, KatsarosD, Tassiulas L (2017) Identifying influential spreaders in complexmultilayer

networks: a centrality perspective. IEEE Trans Netw Sci Eng 6(1):31–45Battiston F, Nicosia V, Latora V (2014) Structural measures for multiplex networks. Phys Rev E

89(3):032804Bayati M, Gleich DF, Saberi A, Wang Y (2013) Message-passing algorithms for sparse network alignment.

ACM Trans Knowl Discov Data 7(1):1–31Bazzi M, Porter MA, Williams S, McDonald M, Fenn DJ, Howison SD (2016) Community detection in

temporal multilayer networks, with an application to correlation networks. Multiscale Model Simul14(1):1–41

BerlingerioM, CosciaM, Giannotti F (2011a) Finding and characterizing communities in multidimensionalnetworks. In: 2011 international conference on advances in social networks analysis andmining. IEEE,pp 490–494

Berlingerio M, Coscia M, Giannotti F (2011b) Finding redundant and complementary communities inmultidimensional networks. In: Proceedings of the 20th ACM international conference on Informationand knowledge management, pp 2181–2184

Berlingerio M, Coscia M, Giannotti F, Monreale A, Pedreschi D (2011c) The pursuit of hubbiness: analysisof hubs in large multidimensional networks. J Comput Sci 2(3):223–237

Berlingerio M, Pinelli F, Calabrese F (2013) Abacus: frequent pattern mining-based community discoveryin multidimensional networks. Data Min Knowl Disc 27(3):294–320

BiondoAE, PluchinoA, RapisardaA (2017) Informative contagion dynamics in amultilayer networkmodelof financial markets. Ital Econ J 3(3):343–366

Black J (2018) Urban transport planning: theory and practice, vol 4. Routledge, AbingdonBlondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large net-

works. J Stat Mech Theory Exp 2008(10):P10008Boccaletti S, Bianconi G, Criado R, Del Genio CI, Gómez-Gardenes J, RomanceM, Sendina-Nadal I,Wang

Z, Zanin M (2014) The structure and dynamics of multilayer networks. Phys Rep 544(1):1–122Bollobás B (2013) Modern graph theory, vol 184. Springer, BerlinBonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Mathe-

matical Sociology 2(1):113–120Bonacich P (1987) Power and centrality: a family of measures. Am J Sociol 92(5):1170–1182Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw ISDN

Syst 30(1–7):107–117Brouwer AE, Haemers WH (2012) Distance-regular graphs. In: Axler S, Capasso V, Casacuberta C, Mac-

Intyre AJ, Ribet K, Sabbah C, Süli E, Woyczynski WA (eds) Spectra of graphs. Springer, pp 177–185Brummitt CD (2014) Models of systemic events: interdependence, contagion, and innovation. University

of California, DavisBuldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of failures in interde-

pendent networks. Nature 464(7291):1025–1028

123

http://arxiv.org/abs/1805.04307

40 X. Huang et al.

Cai D, Shao Z, HeX, Yan X, Han J (2005) Community mining frommulti-relational networks. In: Europeanconference on principles of data mining and knowledge discovery. Springer, pp 445–452

Calimente J (2012) Rail integrated communities in Tokyo. J Trans Land Use 5(1):19–32Cantini L, Medico E, Fortunato S, Caselle M (2015) Detection of gene communities in multi-networks

reveals cancer drivers. Sci Rep 5:17386Caramia M, Dell’Olmo P (2008) Multi-objective management in freight logistics: increasing capacity,

service level and safety with optimization algorithms. Springer, BerlinCarmagnola F, Cena F (2009) User identification for cross-system personalisation. Inf Sci 179(1–2):16–32Carmi S, Havlin S, Kirkpatrick S, Shavitt Y, Shir E (2007) A model of internet topology using k-shell

decomposition. Proc Nat Acad Sci 104(27):11150–11154Cazabet R, Chawuthai R, Takeda H (2015) Using multiple-criteria methods to evaluate community parti-

tions. arXiv preprint arXiv:1502.05149Cellai D, Dorogovtsev SN, Bianconi G (2016) Message passing theory for percolation models on multiplex

networks with link overlap. Phys Rev E 94(3):032301Chen D, Lü L, Shang MS, Zhang YC, Zhou T (2012) Identifying influential nodes in complex networks.

Phys A 391(4):1777–1787Chen D, Huang X, Wang D, Jia L (2014) Public transit hubs identification based on complex networks

theory. IETE Techn Rev 31(6):440–451Chen D, Jia L, Sima D, Huang X, Wang D (2016) Community detection algorithm with membership

function. In: Yuan H, Bian F, Geng J (eds) International conference on geo-informatics in resourcemanagement and sustainable ecosystem. Springer, pp 185–195

Chen J, Li K, Bilal K, Metwally AA, Li K, Yu P (2018a) Parallel protein community detection in large-scalePPI networks based on multi-source learning. IEEE/ACM Trans Comput Biol Bioinform. https://doi.org/10.1109/TCBB.2018.2868088

Chen S, Wang ZZ, Tang L, Tang YN, Gao YY, Li HJ, Xiang J, Zhang Y (2018b) Global vs local modularityfor network community detection. PLoS ONE 13(10):e0205284

Chen Z, Chen C, Zheng Z, Zhu Y (2019) Tensor decomposition for multilayer networks clustering. ProcAAAI Conf Artif Intell 33:3371–3378

Colladon AF, Remondi E (2017) Using social network analysis to prevent money laundering. Expert SystAppl 67:49–58

Cozzo E, Kivelä M, De Domenico M, Solé-Ribalta A, Arenas A, Gómez S, Porter MA, Moreno Y (2015)Structure of triadic relations in multiplex networks. New J Phys 17(7):073029

Cui G, Shrestha R, Han K (2012) Modulesearch: finding functional modules in a protein-protein interactionnetwork. Comput Methods Biomech Biomed Eng 15(7):691–699

Dabideen S, Kawadia V, Nelson SC (2014) Clan: An efficient distributed temporal community detectionprotocol formanets. In: 2014 IEEE 11th international conference onmobile ad hoc and sensor systems.IEEE, pp 91–99

Dalibard V (2012) Community detection in multi-layer networks. Master’s thesis, University of CambridgeDanon L, Diaz-Guilera A, Duch J, Arenas A (2005) Comparing community structure identification. J Stat

Mech Theory Exp 20055(09):P09008Davis A, Gardner BB, Gardner MR (2009) Deep south: a social anthropological study of caste and class.

University of South Carolina Press, ColumbiaDe Bacco C, Power EA, Larremore DB, Moore C (2017) Community detection, link prediction, and layer

interdependence in multilayer networks. Phys Rev E 95(4):042317De Domenico M (2018) Multilayer network modeling of integrated biological systems. arXiv preprint

arXiv:1802.01523De Domenico M, Solé-Ribalta A, Cozzo E, Kivelä M, Moreno Y, Porter MA, Gómez S, Arenas A (2013)

Mathematical formulation of multilayer networks. Phys Rev X 3(4):041022De Domenico M, Nicosia V, Arenas A, Latora V (2014) Layer aggregation and reducibility of multilayer

interconnected networks. arXiv preprint arXiv:1405.0425De Domenico M, Lancichinetti A, Arenas A, Rosvall M (2015a) Identifying modular flows on multilayer

networks reveals highly overlapping organization in interconnected systems. Phys Rev X 5(1):011027De Domenico M, Porter MA, Arenas A (2015b) Muxviz: a tool for multilayer analysis and visualization of

networks. J Compl Netw 3(2):159–176De Domenico M, Solé-Ribalta A, Omodei E, Gómez S, Arenas A (2015c) Ranking in interconnected

multilayer networks reveals versatile nodes. Nat Commun 6(1):1–6

123


https://doi.org/10.1109/TCBB.2018.2868088

https://doi.org/10.1109/TCBB.2018.2868088




De Domenico M, Granell C, Porter MA, Arenas A (2016) The physics of spreading processes in multilayernetworks. Nat Phys 12(10):901–906

DingR,UjangN, binHamidH,AbdMananMS,HeY, Li R,Wu J (2018)Detecting the urban traffic networkstructure dynamics through the growth and analysis of multi-layer networks. Phys A 503:800–817

Drugan OV, Plagemann T, Munthe-Kaas E (2011) Detecting communities in sparse manets. IEEE/ACMTrans Netw 19(5):1434–1447

Du WB, Zhou XL, Jusup M, Wang Z (2016) Physics of transportation: towards optimal capacity using themultilayer network framework. Sci Rep 6:19059

Farina M, Amato P (2002) On the optimal solution definition for many-criteria optimization problems.In: 2002 annual meeting of the North American fuzzy information processing society proceedings.NAFIPS-FLINT 2002 (Cat. No. 02TH8622). IEEE, pp 233–238

Farooq MJ, Zhu Q (2018) On the secure and reconfigurable multi-layer network design for critical infor-mation dissemination in the internet of battlefield things (iobt). IEEE Trans Wireless Commun17(4):2618–2632

Feng S, Wang Q, Shen D, Kou Y, Nie T, Yu G (2017) User identification across social networks based onglobal view features. In: 2017 14th web information systems and applications conference (WISA).IEEE, pp 93–98

Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174Fortunato S, BarthelemyM (2007) Resolution limit in community detection. Proc Nat Acad Sci 104(1):36–

41Fortunato S, Hric D (2016) Community detection in networks: a user guide. Phys Rep 659:1–44Fox PT, Lancaster JL (2002) Mapping context and content: the BrainMap model. Nat Rev Neurosci

3(4):319–321Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40:35–41Freeman LC (1978) Centrality in social networks conceptual clarification. Soc Netw 1(3):215–239Gao J, Buldyrev SV, Havlin S, Stanley HE (2011) Robustness of a network of networks. Phys Rev Lett

107(19):195701Gao X, Zheng Q, Verri FA, Rodrigues RD, Zhao L (2019) Particle competition for multilayer network

community detection. In: Proceedings of the 2019 11th international conference on machine learningand computing, pp 75–80

Girvan M, Newman ME (2002) Community structure in social and biological networks. Proc Nat Acad Sci99(12):7821–7826

Gosak M, Markovic R, Dolenšek J, Rupnik MS, Marhl M, Stožer A, Perc M (2018) Network science ofbiological systems at different scales: a review. Phys Life Rev 24:118–135

Grunwald P (2004) A tutorial introduction to the minimum description length principle. arXiv preprintarXiv:math/0406077

Halu A, Mondragón RJ, Panzarasa P, Bianconi G (2013) Multiplex pagerank. PLoS ONE 8(10):e78293Herlocker JL, Konstan JA, Terveen LG, Riedl JT (2004) Evaluating collaborative filtering recommender

systems. ACM Trans Inf Syst 22(1):5–53Hong C, Liang B (2016) Analysis of the weighted chinese air transportation multilayer network. In: 2016

12th world congress on intelligent control and automation (WCICA). IEEE, pp 2318–2321Huang L, Wang CD, Chao HY (2019) Higher-order multi-layer community detection. Proc AAAI Conf

Artif Intell 33:9945–9946Interdonato R, Tagarelli A, Ienco D, Sallaberry A, Poncelet P (2017) Local community detection in multi-

layer networks. Data Min Knowl Disc 31(5):1444–1479Jeub LG, Mahoney MW, Mucha PJ, Porter MA (2015) A local perspective on community structure in

multilayer networks. arXiv preprint arXiv:1510.05185Jiao P, Lyu H, Li X, YuW,WangW (2017) Temporal community detection based on symmetric nonnegative

matrix factorization. Int J Mod Phys B 31(13):1750102Jutla IS, Jeub LG, Mucha PJ (2011) A generalized louvain method for community detection implemented

in matlab. http://netwikiamathuncedu/GenLouvainKanawati R (2015) Multiplex network mining: A brief survey. IEEE Intell Informatics Bull 16(1):24–27Kao TC, Porter MA (2018) Layer communities in multiplex networks. J Stat Phys 173(3–4):1286–1302Kawadia V, Sreenivasan S (2012) Sequential detection of temporal communities by estrangement confine-

ment. Sci Rep 2:794

123

http://arxiv.org/abs/math/0406077


http://www.netwikiamathuncedu/GenLouvain

42 X. Huang et al.

Kazienko P, Brodka P, Musial K (2010) Individual neighbourhood exploration in complex multi-layeredsocial network. In: 2010 IEEE/WIC/ACM international conference on web intelligence and intelligentagent technology. IEEE, vol 3, pp 5–8

Kempe D, Kleinberg J, Kumar A (2002) Connectivity and inference problems for temporal networks. JComput Syst Sci 64(4):820–842

Kivelä M, Arenas A, Barthelemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. JComplex Netw 2(3):203–271

Kostakos V (2009) Temporal graphs. Phys A 388(6):1007–1023Kuncheva Z, Montana G (2015) Community detection in multiplex networks using locally adaptive random

walks. In: Proceedings of the 2015 IEEE/ACM international conference on advances in social networksanalysis and mining 2015, pp 1308–1315

Kuncheva Z,MontanaG (2017)Multi-scale community detection in temporal networks using spectral graphwavelets. In: International workshop on personal analytics and privacy, Springer, pp 139–154

Kuncheva Z, Montana G (2019) Spectral multi-scale community detection in temporal networks with anapplication. arXiv preprint arXiv:1901.10521

Kurmukov A, Ananyeva M, Dodonova Y, Gutman B, Faskowitz J, Jahanshad N, Thompson P, Zhukov L(2017) Classifying phenotypes based on the community structure of human brain networks. Graphs inbiomedical image analysis, computational anatomy and imaging genetics. Springer, Berlin, pp 3–11

Laird AR, Lancaster JL, Fox PT (2005) The social evolution of a human brain mapping database. Neuroin-formatics 3(1):65–78

Lancichinetti A, Fortunato S (2011) Limits of modularity maximization in community detection. Phys RevE 84(6):066122

Lancichinetti A, Fortunato S, Radicchi F (2008) Benchmark graphs for testing community detection algo-rithms. Phys Rev E 78(4):046110

Lancichinetti A, Fortunato S, Kertész J (2009) Detecting the overlapping and hierarchical communitystructure in complex networks. New J Phys 11(3):033015

Lee DD, Seung HS (2001) Algorithms for non-negative matrix factorization. In: Leen TK, Dietterich TG,Tresp V (eds) Advances in neural information processing systems. MIT Press, pp 556–562. http://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf

Li C, Zhang Y (2020) A personalized recommendation algorithm based on large-scale real micro-blog data.Neural Comput Appl 32:1–8

Li Q, Garcia-Luna-Aceves J (2013) Opportunistic routing using prefix ordering and self-reported socialgroups. In: 2013 international conference on computing, networking and communications (ICNC),IEEE, pp 28–34

Li X, Tian Q, Tang M, Chen X, Yang X (2019) Local community detection for multi-layer mobile networkbased on the trust relation. Wirel Netw 26(8):5503–5515

Li Z, Zhang S, Wang RS, Zhang XS, Chen L (2008) Quantitative function for community detection. PhysRev E 77(3):036109

Liang W, Meng B, He X, Zhang X (2015) Gcm: A greedy-based cross-matching algorithm for identifyingusers across multiple online social networks. In: Pacific-Asia workshop on intelligence and securityinformatics. Springer, pp 51–70

Liu RR, Jia CX, Lai YC (2019) Remote control of cascading dynamics on complex multilayer networks.New J Phys 21(4):045002

Liu W, Suzumura T, Ji H, Hu G (2018) Finding overlapping communities in multilayer networks. PLoSONE 13(4):e0188747

Liu X, Wang W, He D, Jiao P, Jin D, Cannistraci CV (2017) Semi-supervised community detection basedon non-negative matrix factorization with node popularity. Inf Sci 381:304–321

Loe CW, Jensen HJ (2015) Comparison of communities detection algorithms for multiplex. Phys A 431:29–45

Lü L, Zhang YC, Yeung CH, Zhou T (2011) Leaders in social networks, the delicious case. PLoS ONE6(6):e21202

Lü L, Chen D, Ren XL, Zhang QM, Zhang YC, Zhou T (2016) Vital nodes identification in complexnetworks. Phys Rep 650:1–63

Ma X, Dong D, Wang Q (2018) Community detection in multi-layer networks using joint nonnegativematrix factorization. IEEE Trans Knowl Data Eng 31(2):273–286

Ma’ayan A (2017) Complex systems biology. J R Soc Interface 14(134):20170391

123


http://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf

http://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf


MacQueen J et al (1967) Some methods for classification and analysis of multivariate observations. Pro-ceedings of the fifth Berkeley symposium on mathematical statistics and probability, Oakland, CA,USA 1:281–297

Mandaglio D, Amelio A, Tagarelli A (2018) Consensus community detection in multilayer networks usingparameter-free graph pruning. In: Pacific-Asia conference on knowledge discovery and data mining.Springer, pp 193–205

Mondragon RJ, Iacovacci J, Bianconi G (2018) Multilink communities of multiplex networks. PLoS ONE13(3):e0193821

Mucha PJ, Richardson T, Macon K, Porter MA, Onnela JP (2010) Community structure in time-dependent,multiscale, and multiplex networks. Science 328(5980):876–878

Newman M (2018) Networks. Oxford University Press,Newman ME, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E

69(2):026113Newman ME, Leicht EA (2007) Mixture models and exploratory analysis in networks. Proc Nat Acad Sci

104(23):9564–9569NgAY, JordanMI,Weiss Y (2002) On spectral clustering: analysis and an algorithm. In: Advances in neural

information processing systems, pp 849–856Nguyen NP, Dinh TN, Tokala S, Thai MT (2011) Overlapping communities in dynamic networks: their

detection and mobile applications. In: Proceedings of the 17th annual international conference onMobile computing and networking, pp 85–96

Nicolini C, Bifone A (2016) Modular structure of brain functional networks: breaking the resolution limitby surprise. Sci Rep 6(1):1–13

Nicosia V, Latora V (2015) Measuring and modeling correlations in multiplex networks. Phys Rev E92(3):032805

Oselio B, Kulesza A, Hero A (2015) Information extraction from large multi-layer social networks. In:2015 IEEE international conference on acoustics, speech and signal processing (icassp). IEEE, pp5451–5455

PaezMS,AminiAA,LinL (2019)Hierarchical stochastic blockmodel for community detection inmultiplexnetworks. arXiv preprint arXiv:1904.05330

Pamfil AR, Howison SD, Lambiotte R, Porter MA (2019) Relating modularity maximization and stochasticblock models in multilayer networks. SIAM J Math Data Sci 1(4):667–698

Pan Z, Hu G, Li D (2018) Detecting communities from multilayer networks: an aggregation approach. In:proceedings of the international conference on intelligent science and technology, pp 6–11

Paolucci F (2018) Network service chaining using segment routing in multi-layer networks. IEEE/OSA JOpt Commun Netw 10(6):582–592

Papalexakis EE, Akoglu L, Ience D (2013) Do more views of a graph help? community detection andclustering in multi-graphs. In: Proceedings of the 16th international conference on information fusion.IEEE, pp 899–905

Perry JW, Allen K, BerryMM (1955)Machine literature searching x. Machine language; factors underlyingits design and development. Am Doc (pre-1986) 6(4):242

Pizzuti C, Socievole A (2017) Many-objective optimization for community detection in multi-layer net-works. In: 2017 IEEE congress on evolutionary computation (CEC). IEEE, pp 411–418

Pramanik S, Tackx R, Navelkar A, Guillaume JL, Mitra B (2017) Discovering community structure inmultilayer networks. In: 2017 IEEE international conference on data science and advanced analytics(DSAA). IEEE, pp 611–620

Raghavan UN, Albert R, Kumara S (2007) Near linear time algorithm to detect community structures inlarge-scale networks. Phys Rev E 76(3):036106

Rocklin M, Pinar A (2013) On clustering on graphs with multiple edge types. Internet Math 9(1):82–112Roethlisberger FJ, Dickson WJ (2003) Management and the worker, vol 5. Psychology Press, New YorkRossetti G, Guidotti R, Miliou I, Pedreschi D, Giannotti F (2016a) A supervised approach for intra-/inter-

community interaction prediction in dynamic social networks. Soc Netw Anal Mining 6(1):86RossettiG, PappalardoL,RinzivilloS (2016b)Anovel approach to evaluate community detection algorithms

on ground truth. In: Complex networks VII. Springer, pp 133–144Rosvall M, Bergstrom CT (2008) Maps of randomwalks on complex networks reveal community structure.

Proc Nat Acad Sci 105(4):1118–1123Rozario VS, Chowdhury A, Morshed MSJ (2019) Community detection in social network using temporal

data. arXiv preprint arXiv:1904.05291

123



44 X. Huang et al.

Salavati C, Abdollahpouri A, Manbari Z (2018) Bridgerank: a novel fast centrality measure based on localstructure of the network. Phys A 496:635–653

SalehiM, SharmaR,MarzollaM,MagnaniM, Siyari P,Montesi D (2015) Spreading processes inmultilayernetworks. IEEE Trans Netw Sci Eng 2(2):65–83

Sanchez-Rodriguez LM, Iturria-Medina Y, Mouches P, Sotero RC (2019) A method for multiscale commu-nity detection in brain networks. bioRxiv p 743732

Schütze H, Manning CD, Raghavan P (2008) Introduction to information retrieval, vol 39. CambridgeUniversity Press, Cambridge

Silber MD (2011) The Al Qaeda factor: plots against the West. University of Pennsylvania Press, Philadel-phia

Solá L, Romance M, Criado R, Flores J, García del Amo A, Boccaletti S (2013) Eigenvector centrality ofnodes in multiplex networks. Chaos Interdiscip J Nonlinear Sci 23(3):033131

Spirin V, Mirny LA (2003) Protein complexes and functional modules in molecular networks. Proc NatAcad Sci 100(21):12123–12128

Srihari S, YongCH,Wong L (2017) Computational prediction of protein complexes from protein interactionnetworks. Morgan & Claypool, Williston

Sultana S, Salon D, Kuby M (2019) Transportation sustainability in the urban context: a comprehensivereview. Urban Geogr 40(3):279–308

Tagarelli A, Amelio A, Gullo F (2017) Ensemble-based community detection in multilayer networks. DataMin Knowl Disc 31(5):1506–1543

Tang JK, Leontiadis L, Scellato S, et al. (2012a) Temporal network metrics and their application to realworld networks. PhD thesis, Citeseer

Tang L, Wang X, Liu H (2009) Uncoverning groups via heterogeneous interaction analysis. In: 2009 NinthIEEE international conference on data mining. IEEE, pp 503–512

Tang L, Wang X, Liu H (2012b) Community detection via heterogeneous interaction analysis. Data MinKnowl Disc 25(1):1–33

Taylor D, Caceres RS, Mucha PJ (2016a) Detectability of small communities in multilayer and temporalnetworks: eigenvector localization, layer aggregation, and time series discretization. Tech. rep

Taylor D, Shai S, Stanley N,Mucha PJ (2016b) Enhanced detectability of community structure in multilayernetworks through layer aggregation. Phys Rev Lett 116(22):228301

Taylor D, Caceres RS, Mucha PJ (2017) Super-resolution community detection for layer-aggregated mul-tilayer networks. Phys Rev X 7(3):031056

Türker I, Sulak EE (2018) A multilayer network analysis of hashtags in twitter via co-occurrence andsemantic links. Int J Mod Phys B 32(04):1850029

Vaiana M, Muldoon S (2018) Resolution limits for detecting community changes in multilayer networks.arXiv preprint arXiv:1803.03597

Verbrugge LM (1979) Multiplexity in adult friendships. Soc Forces 57(4):1286–1309Wang D, Li J, Xu K, Wu Y (2017) Sentiment community detection: exploring sentiments and relationships

in social networks. Electron Commer Res 17(1):103–132Wang P, Robins G, Pattison P, Lazega E (2013) Exponential random graph models for multilevel networks.

Soc Netw 35(1):96–115Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440–442Wu K, Taki Y, Sato K, Sassa Y, Inoue K, Goto R, Okada K, Kawashima R, He Y, Evans AC et al (2011)

The overlapping community structure of structural brain network in young healthy individuals. PLoSONE 6(5):e19608

Wu W, Kwong S, Zhou Y, Jia Y, Gao W (2018) Nonnegative matrix factorization with mixed hypergraphregularization for community detection. Inf Sci 435:263–281

Xu X, Yuruk N, Feng Z, Schweiger TA (2007) Scan: a structural clustering algorithm for networks. In:Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and datamining, pp 824–833

Yang Y, Yu H, Huang R,Ming T (2018) A fusion information embedding method for user identity matchingacross social networks. In: 2018 IEEE SmartWorld, ubiquitous intelligence & computing, advanced& trusted computing, scalable computing & communications, cloud & big data computing, internetof people and smart city innovation (SmartWorld/SCALCOM/UIC/ATC/CBDCom/IOP/SCI), IEEE,pp 2030–2035

Yiapanis P, Rosas-Ham D, Brown G, Luján M (2013) Optimizing software runtime systems for speculativeparallelization. ACM Trans Archit Code Optim 9(4):1–27

123



Yildirimoglu M, Kim J (2018) Identification of communities in urban mobility networks using multi-layergraphs of network traffic. Transp Res Part C Emerg Technol 89:254–267

Zhan J, Sun M, Wu H, Sun H (2018) Using triangles and latent factor cosine similarity prior to improvecommunity detection in multi-relational social networks. Concurr Comput Practi Exp 30(16):e4453

Zhang F, Ma A, Wang Z, Ma Q, Liu B, Huang L, Wang Y (2018a) A central edge selection based over-lapping community detection algorithm for the detection of overlapping structures in protein-proteininteraction networks. Molecules 23(10):2633

Zhang H, Zhuge C, Yu X (2018b) Identifying hub stations and important lines of bus networks: a case studyin xiamen, china. Phys A 502:394–402

Zhang H, Wang CD, Lai JH, Philip SY (2019) Community detection using multilayer edge mixture model.Knowl Inf Syst 60(2):757–779

Zhang S, Wang RS, Zhang XS (2007) Uncovering fuzzy community structure in complex networks. PhysRev E 76(4):046103

Zhao Y, Karypis G (2004) Empirical and theoretical comparisons of selected criterion functions for docu-ment clustering. Mach Learn 55(3):311–331

Zhou D, Councill I, Zha H, Giles CL (2007) Discovering temporal communities from social networkdocuments. In: Seventh IEEE international conference on data mining (ICDM 2007). IEEE, pp 745–750

Zhou X, Liang W, Wu B, Lu Z, Nishimura S, Shinomiya T, Jin Q (2016) Dynamic community miningand tracking based on temporal social network analysis. In: 2016 IEEE international conference oncomputer and information technology (CIT), IEEE, pp 177–182

Zhu J, Wang B, Wu B, Zhang W (2017) Emotional community detection in social network. IEICE TransInf Syst 100(10):2515–2525

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published mapsand institutional affiliations.

123

a survey of community detection methods in multilayer …...nity detection methods in multilayer...

Documents